Families of lattice polytopes of mixed degree one

TL;DR

This paper classifies families of lattice polytopes with minimal mixed volume properties, showing finiteness in higher dimensions and providing a complete classification in three dimensions, revealing differences across dimensions.

Contribution

It proves a finiteness result for lattice polytope families of mixed degree one in dimensions four and higher, and fully classifies such families in dimension three.

Findings

Finiteness of such families in dimensions ≥4.

Complete classification of these families in dimension 3.

Existence of infinite families in dimension 3 that do not project onto the unimodular triangle.

Abstract

It has been shown by Soprunov that the normalized mixed volume (minus one) of an $n$-tuple of $n$-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined $n$-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension $n \geq 4$, showing that all but finitely many $n$-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex $\Delta_{n-1}$. Furthermore, we give a complete solution in dimension $n=3$. In the course of this we show that our finiteness result does not extend to dimension $n=3$, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle $\Delta_2$.

Equations

$$|(P_1+\cdots +P_n{)}^{\circ }\cap {\mathbb{Z}}^n|\ge \mathrm{MV}(P_1,\dots ,P_n)-1.$$

$$\mathcal{P}(Q):=\mathrm{conv}(Q\times \{\text{0}\}\cup \{{\text{e}}_{n+1}\})\subset {\mathbb{R}}^{n+1}.$$

$$P_1\ast \cdots \ast P_k:=\mathrm{conv}((P_1\times \{\text{0}\})\cup (P_2\times \{{\text{e}}_1\})\cup \cdots \cup (P_k\times \{{\text{e}}_{k-1}\}))\subset {\mathbb{R}}^{n+k-1}.$$

$$Q_k:=$$

$$R_k:=$$

$${\square }_3:=$$

$$\mathrm{conv}({\text{e}}_1,{\text{e}}_2,-{\text{e}}_2)\ast \mathrm{conv}(\text{0},{\text{e}}_1),$$

$$\mathrm{conv}(\text{0},{\text{e}}_1,-{\text{e}}_2)\ast \{-{\text{e}}_2\},$$

$$\mathrm{conv}(\text{0},{\text{e}}_1,{\text{e}}_2)\ast \{{\text{e}}_2\},$$

$$\mathrm{conv}(\text{0},2{\text{e}}_2)\ast \mathrm{conv}(\text{0},{\text{e}}_1),$$

$$\mathrm{conv}(\text{0},-{\text{e}}_1,-{\text{e}}_1-{\text{e}}_2)\ast \{-{\text{e}}_1-2{\text{e}}_2\},$$

$$\mathrm{conv}(\text{0},{\text{e}}_2,-{\text{e}}_1)\ast \{{\text{e}}_1\},$$

$$\mathrm{conv}(\text{0},2{\text{e}}_2)\ast \mathrm{conv}(\text{0},{\text{e}}_1),$$

$$\mathrm{conv}(\text{0},-{\text{e}}_1,-{\text{e}}_1-{\text{e}}_2)\ast \{-{\text{e}}_1-2{\text{e}}_2\},$$

$$\mathrm{conv}(\text{0},-{\text{e}}_2,-{\text{e}}_1)\ast \{{\text{e}}_1-2{\text{e}}_2\}.$$

Full text

1. Introduction

1.1. Basic definitions

A lattice polytope $P\subset{\mathbb{R}}^{n}$ is a polytope $P\subset{\mathbb{R}}^{n}$ whose vertices are elements of the lattice ${\mathbb{Z}}^{n}\subset{\mathbb{R}}^{n}$. We call two lattice polytopes $P_{1},P_{2}\subset{\mathbb{R}}^{n}$ equivalent if there exists an affine lattice-preserving transformation $U\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ satisfying $U(P_{1})=P_{2}$. We say that two $n$-tuples $P_{1},\dots,P_{n}\subset{\mathbb{R}}^{n}$ and $Q_{1},\dots,Q_{n}\subset{\mathbb{R}}^{n}$ are equivalent if there is a permutation $\sigma\in S_{n}$, an affine lattice-preserving transformation $U\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ and vectors $t_{1},\dots,t_{n}\in{\mathbb{Z}}^{n}$ such that $(P_{1},\dots,P_{n})=(U(Q_{\sigma(1)})+t_{1}),\dots,U(Q_{\sigma(n)})+t_{n}))$. We denote by $\Delta_{n}=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\dots,\textnormal{{e}}_{n})\subset{\mathbb{R}}^{n}$ the standard unimodular simplex in ${\mathbb{R}}^{n}$ and call an $n$-dimensional simplex unimodular if it is equivalent to $\Delta_{n}$. We write the Minkowski sum of two lattice polytopes $P_{1},P_{2}\subset{\mathbb{R}}^{n}$ as P_{1}+P_{2}=\mathopen{}\mathclose{{}\left\{{p_{1}+p_{2}\colon p_{1}\in P_{1},p_{2}\in P_{2}}}\right\}\subset{\mathbb{R}}^{n} and denote the interior of a lattice polytope $P\subset{\mathbb{R}}^{n}$ by $P^{\circ}$. If one has $P^{\circ}\cap{\mathbb{Z}}^{n}=\emptyset$, we call the lattice polytope $P\subset{\mathbb{R}}^{n}$ hollow.

1.2. Motivation

In order to give an explicit definition, let us define the (normalized) mixed volume of an $n$-tuple of polytopes $P_{1},\dots,P_{n}\subset{\mathbb{R}}^{n}$ via the inclusion-exclusion formula given by \operatorname{MV}(P_{1},\dots,P_{n})\mathrel{\mathop{:}}=\sum_{\emptyset\neq I\subseteq\mathopen{}\mathclose{{}\left\{{1,\dots,n}}\right\}}(-1)^{n-|I|}{\mathrm{vol}}_{n}(\sum_{i\in I}P_{i}), where ${\mathrm{vol}}_{n}$ denotes the standard euclidean volume in ${\mathbb{R}}^{n}$. Note that there are various equivalent definitions for the mixed volume of an $n$-tuple of lattice polytopes or, more generally, for an $n$-tuple of convex bodies in ${\mathbb{R}}^{n}$ (see e.g. [Sch14] or [EG15]) and that in our definition the mixed volume is normalized such that $\operatorname{MV}(\Delta_{n},\dots,\Delta_{n})=1$. A central connection of the mixed volume to algebraic geometry is given by the famous Bernstein-Kouchnirenko-Khovanskii theorem ([Ber75]). Combining this theorem with a generalization of the Euler-Jacobi theorem due to Khovanskii ([Kho78]) in the context of sparse polynomial interpolation, Soprunov showed the following lower bound on the number of interior lattice points in the Minkowski sum of an $n$-tuple of $n$-dimensional lattice polytopes.

Theorem 1.1** ([Sop07, Nil17]).**

Let $P_{1},\dots,P_{n}\subset{\mathbb{R}}^{n}$ be $n$-dimensional lattice polytopes. Then the following inequality holds:

[TABLE]

Furthermore, equality holds if and only if the Minkowski sum of any choice of $n-1$ polytopes of the tuple $P_{1},\dots,P_{n}$ is hollow.

The $n$-tuples for which equality holds in the above theorem have been called $n$-tuples of mixed degree at most one by Soprunov in [BNR*+*08], where a characterization of such tuples has been posed as a problem ([BNR*+*08, Section 5, Problem 2]). This notion is motivated by a connection to the degree of a lattice polytope, which is an intensively studied invariant in Ehrhart Theory (see for example [BN07, DRP09, DRHNP11, BH18]). The degree $\deg(P)$ of an $n$-dimensional lattice polytope $P\subset{\mathbb{R}}^{n}$ is set to equal $n$ if $P$ has at least one interior lattice point, and otherwise is defined as the smallest integer $0\leq d\leq n-1$ such that the dilated lattice polytope $(n-d)P$ is hollow. Another interpretation is given by the fact that $\deg(P)$ agrees with the degree of the so-called $h^{*}$-polynomial of $P$ (see for example [BR15]). Now in the setting $P_{1}=\dots=P_{n}=P\subset{\mathbb{R}}^{n}$, that is for an $n$-tuple consisting of $n$ copies of the same lattice polytope $P$, the equality condition from Theorem 1.1 is satisfied if and only if the degree of $P$ is at most one. It is a well-known fact that an $n$-dimensional lattice polytope has degree [math] if and only if it is equivalent to $\Delta_{n}$. Also for lattice polytopes of degree one there exists the following complete description by Batyrev-Nill [BN07]. Given an $n$-dimensional lattice polytope $Q\subset{\mathbb{R}}^{n}$, we define the lattice pyramid ${\mathcal{P}}(Q)$ as the $(n+1)$-dimensional polytope

[TABLE]

The lattice pyramid construction preserves the degree (it actually preserves the $h^{*}$-polynomial). We say that an $n$-dimensional lattice polytope is an exceptional simplex if it is equivalent to the polytope obtained via $n-2$ iterations of the lattice pyramid construction over the polygon $2\Delta_{2}$. We say that an $n$-dimensional lattice polytope $P$ is a Lawrence prism if $P$ is equivalent to a lattice polytope $\operatorname{conv}(\{\textnormal{{0}},a_{0}\textnormal{{e}}_{n},\textnormal{{e}}_{1},\textnormal{{e}}_{1}+a_{1}\textnormal{{e}}_{n},\ldots,\textnormal{{e}}_{n-1},\textnormal{{e}}_{n-1}+a_{n-1}\textnormal{{e}}_{n}\})$ for nonnegative integers $a_{0},\ldots,a_{n-1}\in{\mathbb{Z}}_{\geq 0}$.

Theorem 1.2** ([BN07, Theorem 2.5]).**

Let $P$ be a lattice polytope. Then $\deg(P)\leq 1$ (i.e. $(n-1)P$ is hollow) if and only if $P$ is is an exceptional simplex or a Lawrence prism.

The relation of tuples of mixed degree at most one to the degree of a lattice polytope raises the natural question whether there is a general concept of a mixed degree of an $n$-tuple of lattice polytopes in ${\mathbb{R}}^{n}$ that generalizes both Soprunov’s definition of tuples of mixed degree at most one and the degree of a single lattice polytope. A suggestion for such a mixed degree has recently been given by Nill [Nil17]. Let $P_{1},\dots,P_{n}\subset{\mathbb{R}}^{n}$ be an $n$-tuple of lattice polytopes. The mixed degree $\operatorname{md}(P_{1},\dots,P_{n})$ is set to equal $n$ if $P_{i}$ has an interior lattice point for some $1\leq i\leq n$. Otherwise $\operatorname{md}(P_{1},\dots,P_{n})$ is the smallest integer $0\leq d\leq n-1$ such that the Minkowski sum of any choice of $(n-d)$ polytopes of tuple $P_{1},\dots,P_{n}$ is hollow. We refer the reader to [Nil17] for additional motivation for this definition.

In this language, Soprunov’s problem asks for a characterization of $n$-tuples of lattice polytopes $P_{1},\dots,P_{n}\subset{\mathbb{R}}^{n}$ satisfying $\operatorname{md}(P_{1},\dots,P_{n})\leq 1$. The case of $\operatorname{md}(P_{1},\dots,P_{n})=0$ (as this is equivalent to $\operatorname{MV}(P_{1},\dots,P_{n})=1$ by [Nil17, Theorem 2.2]) has already been solved by Cattani et al. in the context of investigating the codimension of so-called mixed discriminants.

Proposition 1.3** ([CCD*+*13, Proposition 2.7]).**

Let $P_{1},\ldots,P_{n}$ be $n$-dimensional lattice polytopes. Then $\operatorname{md}(P_{1},\ldots,P_{n})=0$ if and only if the $n$-tuple $P_{1},\ldots,P_{n}$ is equivalent to the $n$-tuple $\Delta_{n},\dots,\Delta_{n}$.

We therefore often restrict to tuples with mixed degree equal to one in our approach towards solving Soprunov’s problem.

1.3. Results

The contribution of this paper is to partially solve Soprunov’s problem by presenting a finiteness result for dimension $n\geq 4$ and to give a complete characterization of triples of $3$-dimensional lattice polytopes of mixed degree one.

In order to describe a trivial class of $n$-tuples of mixed degree (at most) one, let us introduce the concept of lattice projections. By lattice projection, we denote a surjective affine-linear map $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ satisfying $\varphi({\mathbb{Z}}^{n})={\mathbb{Z}}^{m}$. The kernel of such a projection is affinely generated by lattice points of ${\mathbb{Z}}^{n}$ and we consider two projections to be equal if and only if they have the same kernel up to lattice translations.

The trivial class of $n$-tuples of mixed degree (at most) one is now given by the following example.

Example 1.4**.**

Let $P_{1},\dots,P_{n}\subset{\mathbb{R}}^{n}$ be $n$-dimensional lattice polytopes and $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ a lattice projection satisfying $\varphi(P_{i})=\Delta_{n-1}$ for all $1\leq i\leq n$. Then $P_{1},\ldots,P_{n}$ has mixed degree at most one, as any Minkowski sum of $n-1$ polytopes of the tuple $P_{1},\ldots,P_{n}$ is projected onto the hollow polytope $(n-1)\Delta_{n-1}\subset{\mathbb{R}}^{n-1}$ by $\varphi$. An example of such a trivial tuple in dimension $n=3$ is shown in Figure 1.

One can view $n$-tuples from Example 1.4 as consisting of $n$ lattice polytopes that all are Lawrence prisms and additionally satisfy that they extend into the same height-direction over the same unimodular $(n-1)$-dimensional simplex. Clearly we cannot expect this to be the only class of $n$-tuples of mixed degree one, as already the unmixed setting of Theorem 1.2 additionally yields $n$-tuples of copies of the same exceptional simplex as having mixed degree one. Unlike in the unmixed case there actually exist many more such non-trivial examples (see our classification result for $n=3$ in Theorem 1.6, one example is shown in Figure 2).

This raises the question whether there is any chance to make reasonable statements about $n$-tuples of mixed degree one at all. Our main result is to provide a positive answer to this question by showing that, for any dimension $n$ at least $4$, all but finitely many exceptions of $n$-tuples of mixed degree one are actually of the trivial type described in Example 1.4.

Theorem 1.5**.**

Fix $n\geq 4$ and let $P_{1},\dots,P_{n}\subset{\mathbb{R}}^{n}$ be $n$-dimensional lattice polytopes with $\operatorname{md}(P_{1},\dots,P_{n})=1$. Then, up to equivalence, the $n$-tuple $P_{1},\dots,P_{n}$ either belongs to a finite list of exceptions or there is a lattice projection $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ such that $\varphi(P_{i})=\Delta_{n-1}$ for all $1\leq i\leq n$.

We refer the reader to Section 3 for the proof of Theorem 1.5.

Theorem 1.5 is not true for dimension n\in\mathopen{}\mathclose{{}\left\{{2,3}}\right\}. This fact is straightforward to see for $n=2$, as pairs of lattice polygons $P_{1},P_{2}\subset{\mathbb{R}}^{2}$ are of mixed degree (at most) one if and only if both $P_{1}$ and $P_{2}$ are hollow. Fixing $P_{1}$ to be any hollow polygon and letting $P_{2}$ range through all polygons that are equivalent to a fixed hollow polygon will clearly yield infinitely many non-equivalent pairs of mixed degree one without there being a projection commonly mapping both polytopes onto the segment $\Delta_{1}$.

For $n=3$, however, we find that only a very specific class of triples of mixed degree one contains an infinite number of exceptions and we can explicitly describe a finite number of $1$-parameter families covering this class. This is part of the following classification result, which essentially gives a complete answer to Soprunov’s problem for dimension $n=3$. Note that we say that a $k$-tuple of $n$-dimensional lattice polytopes $P_{1},\dots,P_{k}$ admits a common lattice projection onto translates of an $(n-1)$-dimensional lattice polytope $Q$ if there exists a lattice projection $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ satisfying $\varphi(P_{i})=Q+t_{i}$ for all $1\leq i\leq k$ and some $t_{i}\in{\mathbb{Z}}^{n-1}$.

Theorem 1.6**.**

Let $P_{1},P_{2},P_{3}\subset{\mathbb{R}}^{3}$ be a triple of $3$-dimensional lattice polytopes that satisfies $\operatorname{md}(P_{1},P_{2},P_{3})=1$. Then either there is a lattice projection $\varphi\colon{\mathbb{R}}^{3}\to{\mathbb{R}}^{2}$ such that $\varphi(P_{i})=\Delta_{2}$ for all $1\leq i\leq 3$, or one of the following holds.

1. (i)

There is no pair in $P_{1},P_{2},P_{3}$ admitting a common lattice projection onto translates of $\Delta_{2}$ and $P_{1},P_{2},P_{3}$ is equivalent to one out of $29$ possible triples. 2. (ii)

There is exactly one pair in $P_{1},P_{2},P_{3}$ admitting a common lattice projection onto translates of $\Delta_{2}$ and $P_{1},P_{2},P_{3}$ is equivalent to one out of $141$ possible triples. 3. (iii)

There are exactly two pairs in $P_{1},P_{2},P_{3}$ admitting a common lattice projection onto translates of $\Delta_{2}$ and $P_{1},P_{2},P_{3}$ is equivalent to one out of $82$ possible triples. 4. (iv)

All pairs in $P_{1},P_{2},P_{3}$ admit a common lattice projection onto translates of $\Delta_{2}$ and

1. (a)

the kernels of the projections cannot be shifted into a common hyperplane and the triple $P_{1},P_{2},P_{3}$ is equivalent to one out of $27$ possible triples. 2. (b)

the kernels of the projections can be shifted into a common hyperplane and $P_{1},P_{2},P_{3}$ belongs, up to equivalence, to one out of finitely many infinite 1-parameter families of triples.

We refer the reader to Section 4 for a proof of Theorem 1.6.

In the following example we present one of the $1$-parameter families from Theorem 1.6 (iv). Let us denote $\square_{2}\mathrel{\mathop{:}}=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2})\subset{\mathbb{R}}^{2}$.

Example 1.7**.**

Let $P^{k}_{1},P^{k}_{2},P_{3}\subset{\mathbb{R}}^{3}$ be the triple given by

• $P^{k}_{1}\mathrel{\mathop{:}}=\operatorname{conv}(\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},2\textnormal{{e}}_{2},k\textnormal{{e}}_{2}+\textnormal{{e}}_{3})$,
• $P^{k}_{2}\mathrel{\mathop{:}}=\operatorname{conv}(\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},2\textnormal{{e}}_{1},k\textnormal{{e}}_{1}+\textnormal{{e}}_{3})$,
• $P_{3}\mathrel{\mathop{:}}=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},\textnormal{{e}}_{3})$,

for some $k\in{\mathbb{Z}}_{\geq 0}$. Then $\operatorname{md}(P^{k}_{1},P^{k}_{2},P_{3})=1$ and, while all pairs in $P^{k}_{1},P^{k}_{2},P_{3}$ admit a common lattice projection onto translates of $\Delta_{2}$, there is no lattice projection commonly mapping the whole triple $P^{k}_{1},P^{k}_{2},P_{3}$ onto translates of $\Delta_{2}$. Note that $P^{k}_{1},P^{k}_{2}$ and $P_{3}$ as single lattice polytopes are all equivalent to ${\mathcal{P}}(\square_{2})$ for all $k\in{\mathbb{Z}}_{\geq 0}$ (see Figure 3).

All computations have been carried out using Magma [BCP97] and the code can be found at https://github.com/gabrieleballetti/mixed_degree_one.

Acknowledgements

We thank Benjamin Nill for providing the initial idea for the project and giving helpful advice and comments throughout the progress. Furthermore we thank Ivan Soprunov for valuable feedback on an intermediate version of the paper.

We also thank Akiyoshi Tsuchiya and Takayuki Hibi for the opportunity to work together on the project during the Summer School and Workshop on Lattice Polytopes in Osaka 2019. This article can be seen as a vast generalization of our first investigation on the 3-dimensional case that appeared in the note “Families of 3-dimensional polytopes of mixed degree one” in the proceedings of that conference ([HT19]).

The first author was partially supported by the Vetenskapsrådet grant NT:2014-3991. The second author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 314838170, GRK 2297 MathCoRe.

2. Cayley sums and projections

While there is a well-known way of determining whether two lattice polytopes are equivalent by comparing their normal forms (see e.g. [KS98]), the task of checking two tuples for equivalence seems a priori more complicated. However, we will show that in our setting the construction of the Cayley sum of a tuple allows us to reduce the problem of checking equivalence of tuples to checking equivalence of two higher-dimensional lattice polytopes. The Cayley sum (or Cayley polytope) construction occurs in various contexts in the literature, for example in the construction of mixed subdivisions of Minkowski sums ([DLRS10]), in the study of $A$-discriminants ([FI16, GKZ94]) and in structural results on lattice polytopes of high dimension and small degree ([DRHNP11, DRP09]).

Let us recall the definition and some basic properties of the Cayley sum of a tuple of lattice polytopes.

Definition 2.1**.**

Let $P_{1},\dotsc,P_{k}\subset{\mathbb{R}}^{n}$ be lattice polytopes. We define the Cayley sum $P_{1}*\dots*P_{k}$ as

[TABLE]

We call a Cayley sum $P_{1}*\dots*P_{k}$ proper if $P_{i}\neq\emptyset$ for all $1\leq i\leq k$. In this case one has $\dim(P_{1}*\dots*P_{k})=\dim(P_{1}+\dots+P_{k})+k-1$.

Proposition 2.2** ([BN08, Proposition 2.3]).**

Let $P\subset{\mathbb{R}}^{n}$ be a lattice polytope. Then the following are equivalent.

1. (i)

There exists a lattice projection $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{k-1}$ with $\varphi(P)=\Delta_{k-1}$, 2. (ii)

there exist lattice polytopes $P_{1},\dots,P_{k}\subset{\mathbb{R}}^{n-k+1}$ such that $P\cong P_{1}*\dots*P_{k}$.

In particular if $k=n$, $P$ is the Cayley sum of $n$ segments.

Remark 2.3**.**

Let $P=P_{1}*\dots*P_{k}$. Then all faces of $P$ are of the form $F=F_{1}*\dots*F_{k}$ for (possibly empty) faces $F_{i}\subseteq P_{i}$. One has $\dim(F)=\dim(F_{1}+\dots+F_{k})+l-1$ where $l$ is the number of $F_{i}$ that satisfy $F_{i}\neq\emptyset$. In particular, each of the $P_{i}$ corresponds to a face of $P$ which we will denote by $\hat{P_{i}}$. Furthermore $\hat{P}_{i}^{c}$, by which we denote the convex hull of all vertices of $P$ that are not contained in $\hat{P_{i}}$, is always a proper face of $P$.

Let us now formulate the lemma which allows us to determine equivalence of certain $n$-dimensional tuples by comparing the normal forms of their Cayley sums.

Lemma 2.4**.**

Let $P_{1},\dots,P_{k}$ and $Q_{1},\dots,Q_{k}$ be $k$-tuples of $n$-dimensional lattice polytopes in ${\mathbb{R}}^{n}$. Assume that there is no lattice projection $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{k-1}$ mapping $P_{1},\dots,P_{k}$ onto translates of $\Delta_{k-1}$. Then the $k$-tuples $P_{1},\dots,P_{k}$ and $Q_{1},\dots,Q_{k}$ are equivalent if and only if their Cayley sums $P_{1}*\dots*P_{k}$ and $Q_{1}*\dots*Q_{k}$ are equivalent.

Proof.

The fact that Cayley sums of equivalent $k$-tuples of lattice polytopes are equivalent is a straightforward consequence of the definition. Suppose now that the $k$-tuple $P_{1},\dots,P_{k}$ does not admit a common projection onto translates of $\Delta_{k-1}$ and suppose that $P_{1}*\dots*P_{k}$ and $Q_{1}*\dots*Q_{k}$ are equivalent. Let $P_{1}^{\prime},\dots,P_{k}^{\prime}$ be the images of $\hat{P}_{1},\dots,\hat{P}_{k}$ under the lattice-preserving transformation yielding the equivalence, respectively. We will first show that, up to renumbering, $P_{i}^{\prime}=\hat{Q}_{i}$ for all $1\leq i\leq k$. Suppose without loss of generality $P_{1}^{\prime}\neq\hat{Q}_{i}$ for all $1\leq i\leq k$. Note that $P_{1}^{\prime}$ cannot properly contain any $\hat{Q}_{i}$ (by Remark 2.3 any face of $Q_{1}*\dots*Q_{k}$ that properly contains a $\hat{Q}_{i}$ is of dimension greater or equal to $n+1$). This also implies that $P_{1}^{\prime}$ cannot be disjoint to some $\hat{Q}_{i}$ due to the following. If $P_{1}^{\prime}$ was disjoint to, say, $\hat{Q}_{1}$, its complement $(P_{1}^{\prime})^{c}$ would contain $\hat{Q}_{1}$. As however $P_{1}^{\prime}$ does not contain any $\hat{Q}_{i}$, the complement $(P_{1}^{\prime})^{c}$ contains at least one point of $\hat{Q}_{i}$ for each $1\leq i\leq k$. Therefore, if we pick points $p_{2},\dots,p_{k}$ in $\hat{Q}_{2}\setminus P_{1}^{\prime},\dots,\hat{Q}_{k}\setminus P_{1}^{\prime}$ respectively, we have \dim(\hat{P}_{1}^{c})=\dim((P_{1}^{\prime})^{c})\geq\dim(\hat{Q}_{1}*\mathopen{}\mathclose{{}\left\{{p_{2}}}\right\}*\dots*\mathopen{}\mathclose{{}\left\{{p_{k}}}\right\})=n+k-1, which is a contradiction to $\hat{P_{1}^{c}}$ being a proper face of the Cayley sum $P_{1}*\dots*P_{k}$ (see Remark 2.3).

As $P_{1}^{\prime}$ was chosen arbitrarily this argumentation implies that all $P_{i}^{\prime}$ have non-empty intersection with all $\hat{Q}_{j}$. Therefore the natural projection of $Q_{1}*\dots*Q_{k}$ onto $\Delta_{k-1}$ remains surjective when restricted to the affine hull of $P_{i}^{\prime}$ for each $1\leq i\leq k$. As the affine hulls of $P^{\prime}_{1},\dots,P^{\prime}_{k}$ are by construction pairwise parallel $n$-dimensional affine subspaces of ${\mathbb{R}}^{n+k-1}$ the natural projection yields a lattice projection $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{k-1}$ mapping $P_{i}$ onto a translate of $\Delta_{k-1}$ for each $1\leq i\leq k$. This contradicts our assumption.

This shows that, if $P_{1},\dots,P_{k}$ do not have a common projection onto translates of $\Delta_{k-1}$, any affine lattice-preserving transformation $U\colon{\mathbb{R}}^{n+k-1}\to{\mathbb{R}}^{n+k-1}$ mapping $P_{1}*\dots*P_{k}$ to $Q_{1}*\dots*Q_{k}$ yields (up to renumbering) a bijection mapping the face $\hat{P}_{i}$ to the face $\hat{Q}_{i}$. We may without loss of generality assume that $U$ preserves the origin. Restricting $U$ to a map from the affine hull of $P_{1}$ to the affine hull of $Q_{1}$ we obtain a linear lattice-preserving transformation $L\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$. It is straightforward to verify that restricting $U$ to the affine hull of any other $P_{i}$ results in a map $x\mapsto L(x)+t_{i}$ from ${\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ for some $t_{i}\in{\mathbb{Z}}^{n}$. ∎

3. Proof of the main theorem

From Proposition 2.2 we can easily deduce the following lemma. We call a facet $F$ of a lattice polytope $P\subset{\mathbb{R}}^{n}$ unimodular if $F$ is a unimodular simplex inside the affine lattice defined as the intersection of the affine hull of $F$ with ${\mathbb{Z}}^{n}$.

Lemma 3.1**.**

Let $P\subset{\mathbb{R}}^{n}$ be an $n$-dimensional lattice polytope and $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ a lattice projection that projects $P$ onto $\Delta_{n-1}$. Then $\ker\varphi={\mathbb{R}}e$ where $e$ is a vector parallel to an edge between vertices $v_{1},v_{2}$, where $v_{1}\in F_{1}$ and $v_{2}\in F_{2}$ for two different unimodular facets $F_{1}\neq F_{2}$ of $P$.

We now study $n$-dimensional polytopes projecting onto $\Delta_{n-1}$ along multiple directions. Recall that we denote by $\mathcal{P}^{n-2}(\square_{2})$ the $(n-2)$-fold lattice pyramid formed over the square $\square_{2}=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2})\subset{\mathbb{R}}^{2}$.

Lemma 3.2**.**

Let $P\subset{\mathbb{R}}^{n}$ be an $n$-dimensional lattice polytope such that there are different lattice projections $\varphi_{1},\varphi_{2}\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ that map $P$ onto $\Delta_{n-1}$. Then $P$ is equivalent either to the unimodular simplex $\Delta_{n}$ or to $\mathcal{P}^{n-2}(\square_{2})$. If there exists another projection $\varphi_{3}\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ mapping $P$ onto $\Delta_{n-1}$, then $P$ is necessarily equivalent to $\Delta_{n}$.

Proof.

As $P$ has one projection onto $\Delta_{n-1}$, by Proposition 2.2 we may assume that $P$ is of the form $P=I_{1}*\dots*I_{n}$ for $n$ segments $I_{i}=[0,a_{i}]$ with $a_{i}\in{\mathbb{Z}}_{\geq 0}$. Two facets of $P$ are given by \Delta_{n-1}\times\mathopen{}\mathclose{{}\left\{{0}}\right\} and \mathopen{}\mathclose{{}\left\{{a_{1}}}\right\}*\dots*\mathopen{}\mathclose{{}\left\{{a_{n}}}\right\}. All other facets of $P$ are of a form we denote by $F_{k}$ for $1\leq k\leq n$, which is the Cayley sum of all $I_{i}$ excluding $I_{k}$. As there exists another lattice projection $\varphi_{2}$ mapping $P$ onto $\Delta_{n-1}$, by Lemma 3.1 the facet $F_{k}$ has to be unimodular for some $1\leq k\leq n$. Assume without loss of generality that $F_{1}$ is unimodular and $a_{2}=1$ and $a_{3}=\dots=a_{n}=0$. Furthermore, $a_{1}$ cannot be greater than one as otherwise $P$ would have an edge of lattice length at least $2$. Therefore any projection which is not along this edge direction could not be projecting $P$ onto $\Delta_{n-1}$. If $a_{1}=0$, then $P$ is equivalent to $\Delta_{n}$, otherwise $a_{1}=1$ and $P$ is equivalent to $\mathcal{P}^{n-2}(\square_{2})$. One easily verifies that $\mathcal{P}^{n-2}(\square_{2})$ does not have more than two different projections onto $\Delta_{n-1}$. ∎

Lemma 3.3**.**

Let $S_{1},S_{2}\subset{\mathbb{R}}^{n}$ be two unimodular $n$-dimensional simplices, $u_{1},u_{2}\in{\mathbb{Z}}^{n}$ be linearly independent edge directions for $S_{1}$ and $S_{2}$ respectively, and $C_{1},C_{2}\subset{\mathbb{R}}^{n}$ be the infinite prisms $S_{1}+{\mathbb{R}}u_{1}$ and $S_{2}+{\mathbb{R}}u_{2}$ respectively. Given $z\in{\mathbb{Z}}^{n}$, denote by $P_{z}$ the intersection $\operatorname{conv}(C_{1}\cap(C_{2}+z)\cap{\mathbb{Z}}^{n})$. Let $v,w\in{\mathbb{Z}}^{n}$ such that $P_{v}$ and $P_{w}$ are both $n$-dimensional. Then $P_{v}$ and $P_{w}$ are the same lattice polytope up to translation.

Proof.

By Lemma 3.2 we know that, if there exists $v\in{\mathbb{Z}}^{n}$ such that $P_{v}$ is $n$-dimensional, then $P_{v}$ is equivalent either to $\Delta_{n}$ or to $\mathcal{P}^{n-2}(\square_{2})$ having two edges parallel to the directions $u_{1}$ and $u_{2}$. In either of the two cases, we can assume $P_{v}$ to be exactly $\Delta_{n}$ or $\mathcal{P}^{n-2}(\square_{2})$.

If $P_{v}=\mathcal{P}^{n-2}(\square_{2})$ then, up to reordering and changes of signs, $u_{1}=\textnormal{{e}}_{1}$ and $u_{2}=\textnormal{{e}}_{2}$. In particular, $C_{1}=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{2},\ldots,\textnormal{{e}}_{n})+{\mathbb{R}}\textnormal{{e}}_{1}$ and $C_{2}+v=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{3},\ldots,\textnormal{{e}}_{n})+{\mathbb{R}}\textnormal{{e}}_{2}$. One easily verifies that $C_{1}\cap(C_{2}+w)$ is full-dimensional if and only if $w-v\in{\mathbb{Z}}\textnormal{{e}}_{1}+{\mathbb{Z}}\textnormal{{e}}_{2}$. In all these cases $C_{1}\cap(C_{2}+w)$ is a translation of $P_{v}$.

On the other hand, if $P_{v}=\Delta_{n}$, then there is another case distinction. If $u_{1}$ and $u_{2}$ are parallel to adjacent edges of $D_{n}$, then we can assume $u_{1}=\textnormal{{e}}_{1}$ and $u_{2}=\textnormal{{e}}_{2}$. But in this case $C_{1}$ and $C_{2}+v$ must intersect in $\mathcal{P}^{n-2}(\square_{2})$ instead of in $\Delta_{n}$, hence we have a contradiction. Therefore $u_{1}$ and $u_{2}$ are parallel to non-adjacent edges of $\Delta_{n}$ and we can assume $u_{1}=\textnormal{{e}}_{1}$ and $u_{2}=\textnormal{{e}}_{2}-\textnormal{{e}}_{3}$. In particular, $C_{1}=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{2},\ldots,\textnormal{{e}}_{n})+{\mathbb{R}}\textnormal{{e}}_{1}$ and $C_{2}+v=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{3},\ldots,\textnormal{{e}}_{n})+{\mathbb{R}}(\textnormal{{e}}_{2}-\textnormal{{e}}_{3})$. Again, one easily verifies that $C_{1}\cap(C_{2}+w)$ is full-dimensional if and only if $w-v\in{\mathbb{Z}}\textnormal{{e}}_{1}+{\mathbb{Z}}(\textnormal{{e}}_{2}-\textnormal{{e}}_{3})$. In all these cases $C_{1}\cap(C_{2}+w)$ is a translation of $\Delta_{n}$. ∎

Lemma 3.4**.**

Let $P\subset{\mathbb{R}}^{n}$ be a unimodular $n$-simplex and $\varphi_{1},\varphi_{2}\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ be two different lattice projections such that, for each $1\leq i\leq 2$, the images $\varphi_{i}(P)$ and $\varphi_{i}(\Delta_{n})$ are translates of $\Delta_{n-1}$. Then, up to translation and coordinate permutation, $P$ is contained in $\mathcal{P}^{n-2}(\square_{2})$. If there exists another projection $\varphi_{3}\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ mapping $P$ and $\Delta_{n}$ onto translates of $\Delta_{n-1}$, then $P$ is necessarily a translate of $\Delta_{n}$.

Proof.

By Lemma 3.1, $\varphi_{1}$ and $\varphi_{2}$ are projections along the directions $u_{1}$ and $u_{2}$ of two edges of $\Delta_{n}$. If $u_{1}$ and $u_{2}$ are the directions of two adjacent edges of $\Delta_{n}$, we can suppose that $u_{1}=\textnormal{{e}}_{1}$ and $u_{2}=\textnormal{{e}}_{2}$. Then $P$ is contained in the intersection $(C_{1}+z_{1})\cap(C_{2}+z_{2})$ where $C_{1}\mathrel{\mathop{:}}=\Delta_{n}+{\mathbb{R}}\textnormal{{e}}_{1}$ and $C_{2}\mathrel{\mathop{:}}=\Delta_{n}+{\mathbb{R}}\textnormal{{e}}_{2}$, for some $z_{1},z_{2}\in{\mathbb{Z}}^{n}$. By Lemma 3.3, $P$ is, up to translation, contained in $C_{1}\cap C_{2}=\mathcal{P}^{n-2}(\square_{2})$. If $u_{1}$ and $u_{2}$ are the directions of two non-adjacent edges of $\Delta_{n}$ then we can suppose that $u_{1}=\textnormal{{e}}_{1}$ and $u_{2}=\textnormal{{e}}_{2}-\textnormal{{e}}_{3}$. Then $P$ is contained in the intersection $(C_{1}+z_{1})\cap(C_{2}+z_{2})$ where $C_{1}\mathrel{\mathop{:}}=\Delta_{n}+{\mathbb{R}}\textnormal{{e}}_{1}$ and $C_{2}\mathrel{\mathop{:}}=\Delta_{n}+{\mathbb{R}}(\textnormal{{e}}_{2}-\textnormal{{e}}_{3})$, for some $z_{1},z_{2}\in{\mathbb{Z}}^{n}$. By Lemma 3.3, $P$ is, up to translation, contained in $C_{1}\cap C_{2}=\Delta_{n}$, therefore $P$ is a translate of $\Delta_{n}\subset\mathcal{P}^{n-2}(\square_{2})$. This proves the first part of the statement.

For the second part of the statement we note that $\varphi_{3}$ must also be a projection along the direction $u_{3}$ of an edge of $\Delta_{n}$. The only case we need to check is when the edges parallel to $u_{1}$, $u_{2}$ and $u_{3}$ form a triangle in $\Delta_{n}$. Indeed, if this is not the case either two of these edges are non-adjacent and $P$ must be a translate of $\Delta_{n}$ as above, or $u_{1}$,$u_{2}$ and $u_{3}$ share a vertex. In the latter case we may assume $u_{i}=\textnormal{{e}}_{i}$ for $1\leq i\leq 3$. As deduced above from Lemma 3.3, this in particular yields that $P$ is contained in the intersection of a translation of the square pyramid $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},\textnormal{{e}}_{3},\dots,\textnormal{{e}}_{n})$ with the flipped square pyramid $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{3},\textnormal{{e}}_{1}+\textnormal{{e}}_{3},\textnormal{{e}}_{2},\textnormal{{e}}_{4},\dots,\textnormal{{e}}_{n})$. This implies that $P$ is a translate of $\Delta_{n}$. Let us therefore assume that $u_{1}=\textnormal{{e}}_{1}$, $u_{2}=\textnormal{{e}}_{2}$ and $u_{3}=\textnormal{{e}}_{1}-\textnormal{{e}}_{2}$. In this case $P$ is a translate of one of the four $n$-dimensional subpolytopes of $\mathcal{P}^{n-2}(\square_{2})$. It is easy to verify that $\Delta_{n}$ is the only one of them that is projected by $\varphi_{3}$ onto a translate of $\varphi_{3}(\Delta_{n})$. ∎

Definition 3.5**.**

Let $P_{1},\dots,P_{n-1}\subset{\mathbb{R}}^{n}$ be $n$-dimensional polytopes with the Minkowski sum $P_{1}+\dots+P_{n-1}$ being hollow. We call the $(n-1)$-tuple $P_{1},\dots,P_{n-1}$ exceptional, if there exists no projection $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ such that $\varphi(P_{1}+\dots+P_{n-1})\subset{\mathbb{R}}^{n-1}$ is a hollow polytope.

Remark 3.6**.**

By [NZ11, Theorem 1.2] there exist only finitely many $n$-dimensional lattice polytopes not admitting a lattice projection onto a hollow $(n-1)$-dimensional lattice polytope, up to equivalence. So in particular, up to equivalence, there exist only finitely many exceptional $(n-1)$-tuples of $n$-dimensional lattice polytopes.

Furthermore, by Proposition 1.3, for any non-exceptional $(n-1)$-tuple $P_{1},\dots,P_{n-1}$ of $n$-dimensional lattice polytopes there exists a lattice projection $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ mapping all $P_{i}$ onto translates of $\Delta_{n-1}$.

Proof of Theorem 1.5.

Let $n\geq 4$. Given $1\leq k\leq n$, denote by $I_{k}$ the set \mathopen{}\mathclose{{}\left\{{1,...,n}}\right\}\setminus\{k\}, and by $[P]_{k}$ the $(n-1)$-tuple given by all $P_{i}$ for $i\in I_{k}$. Denote furthermore by $P_{I_{k}}$ the Minkowski sum $\sum_{i\in I_{k}}P_{i}$ of the polytopes in $[P]_{k}$. Since $\operatorname{md}(P_{1},\dots,P_{n})=1$, the Minkowski sum $P_{I_{k}}$ is hollow for any $1\leq k\leq n$. Recall that, if $[P]_{k}$ is not exceptional, then by Remark 3.6 there exists a projection $\varphi:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ mapping all polytopes in $[P]_{k}$ onto translates of $\Delta_{n-1}$. We treat cases separately, depending on the number of exceptional $(n-1)$-subtuples of the tuple $P_{1},\ldots,P_{n}$.

1. (0)

If $P_{1},\ldots,P_{n}$ has no exceptional $(n-1)$-subtuples then either there exists a projection $\varphi:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ mapping $P_{1},\ldots,P_{n}$ onto translates of $\Delta_{n-1}$ (and in this case there is nothing to prove), or each of the $P_{i}$ admits $n-1$ pairwise different projections onto $\Delta_{n-1}$. Indeed if two of these projections were the same, then we would be in the previous case. Suppose there exist $n-1$ pairwise different projections. As $n\geq 4$, Lemma 3.2 yields that each of the $P_{i}$ is a unimodular $n$-dimensional simplex. Without loss of generality we assume $P_{1}=\Delta_{n}$. Given $2\leq i\leq n$, there exist $n-2$ pairwise different projections mapping $P_{1}$ and $P_{i}$ onto translates of $\Delta_{n-1}$. If $n\geq 5$, by Lemma 3.4, we can immediately deduce that, up to translations, $P_{1}=P_{2}=\ldots=P_{n}=\Delta_{n}$. If $n=4$, Lemma 3.4 only ensures that $P_{2},\ldots,P_{n}$ are, up to translation and coordinate permutation, contained in $\mathcal{P}^{n-2}(\square_{2})$. This yields finitely many cases and checking them computationally we find among them only $4$-tuples admitting a common projection onto $\Delta_{3}$. 2. (1)

$P_{1},\ldots,P_{n}$ has exactly one exceptional $(n-1)$-subtuple, which we can assume to be $[P]_{n}$. As $[P]_{n}$ is an exceptional $(n-1)$-tuple, the Minkowski sum $P_{I_{n}}$ belongs to a finite list of hollow $n$-dimensional polytopes. This means that there are, up to equivalence, finitely many exceptional tuples to choose $[P]_{n}$ from. We now show, that given $[P]_{n}$ there are finitely many possible choices for $P_{n}$ that lead to the $n$-tuple $P_{1},\dots,P_{n}$ having exactly $[P]_{n}$ as an exceptional $(n-1)$-subtuple, which shows the finiteness of this case.

Let therefore $\varphi_{1}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ be a lattice projection mapping the lattice polytopes in $[P]_{2}$ to translates of $\Delta_{n-1}$. Similarly, let $\varphi_{2}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ be a lattice projection mapping the lattice polytopes in $[P]_{1}$ to translates of $\Delta_{n-1}$. The existence of such projections follows from the fact that $[P]_{2}$ and $[P]_{1}$ are non-exceptional. We remark that there exist finitely many such projections. Let $C_{i}$ be the infinite prism $P_{i}+\ker\varphi_{i}$, for $1\leq i\leq 2$. Then we know that any possible choice of $P_{n}$ is contained in $(C_{1}+z_{1})\cap(C_{2}+z_{2})$ for some $z_{1},z_{2}\in{\mathbb{Z}}^{n}$. By Lemma 3.3, for any choices of lattice points $z_{1},z_{2},z_{1}^{\prime},z_{2}^{\prime}\in{\mathbb{Z}}^{n}$ such that $\dim((C_{1}+z_{1})\cap(C_{2}+z_{2})\cap{\mathbb{Z}}^{n})=\dim((C_{1}+z_{1}^{\prime})\cap(C_{2}+z_{2}^{\prime})\cap{\mathbb{Z}}^{n})=n$ we find that $(C_{1}+z_{1}^{\prime})\cap(C_{2}+z_{2}^{\prime})$ is a translate of $(C_{1}+z_{1})\cap(C_{2}+z_{2})$. Therefore, up to translations, all possible choices for $P_{n}$ are contained in $(C_{1}+z_{1})\cap(C_{2}+z_{2})$ for fixed $z_{1},z_{2}\in{\mathbb{Z}}^{n}$. Note that the intersection $(C_{1}+z_{1})\cap(C_{2}+z_{2})$ is either equivalent to $\Delta_{n}$ or $\mathcal{P}^{n-2}(\square_{2})$ by Lemma 3.2, where the choice of the equivalence class depends entirely on $[P]_{n}$. This implies that $P_{n}$ must be one element of a finite list of lattice polytopes fully determined by $[P]_{n}$. 3. (2+)

If $P_{1},\ldots,P_{n}$ has two or more exceptional $(n-1)$-subtuples, then we can suppose that $[P]_{n}$ and $[P]_{n-1}$ are exceptional. In particular, there exists an upper bound depending only on $n$ for the volume of the Minkowski sums $P_{I_{n}}$ and $P_{I_{n-1}}$ and therefore (since $n>2$) for the volume of $P_{1}+P_{i}$ for any $2\leq i\leq n$. Recall that, by [LZ91, Theorem 2], there are, up to equivalence, only finitely many lattice polytopes of any fixed volume $K\in{\mathbb{Z}}_{\geq 0}$. Therefore, as in particular the volume of $P_{1}$ is bounded, there exist only finitely many choices for $P_{1}$ up to equivalence. Furthermore, fixing $P_{1}$ determines, up to translation, finitely many possibilities for each $P_{i}$ with $2\leq i\leq n$ due to the volume bound on $P_{1}+P_{i}$. This yields that there are only finitely many $n$-tuples $P_{1},\dots,P_{n}$ in this case, up to equivalence.

∎

Note, that the assumption $n>3$ is only used in case $(0)$ of the previous proof.

The unmixed result of Theorem 1.2 also gives an explicit description of lattice polytopes of degree one that are not Lawrence prisms, in fact, up to equivalence and the lattice pyramid construction, there exists only one such exception over all dimensions. Such an explicit description of the list of exceptions from the statement of Theorem 1.5 is not known in dimension $n\geq 4$.

Question 3.7**.**

For dimension $n\geq 4$, what are the $n$-tuples of $n$-dimensional lattice polytopes $P_{1},\dots,P_{n}\subset{\mathbb{R}}^{n}$ with $\operatorname{md}(P_{1},\dots,P_{n})=1$ that are not of the trivial type described in Example 1.4? Is there a finite description over all dimensions as there is in the unmixed case?

In [Nil17] the mixed degree is actually treated in a more general way, also being defined for $m$-tuples of $n$-dimensional lattice polytopes, with $m\neq n$. In particular, an $m$-tuple of lattice polytopes $P_{1},\dots,P_{m}\subset{\mathbb{R}}^{n}$ satisfies $\operatorname{md}(P_{1},\dots,P_{m})\leq 1$ if and only if $m\geq n-1$ and the Minkowski sum of each $(n-1)$-subtuple is hollow. For $m=n-1$ we obtain an analogous result to Theorem 1.5 (even for n\in\mathopen{}\mathclose{{}\left\{{2,3}}\right\}) immediately from [NZ11, Theorem 1.2]. We remark that Theorem 1.5 also inductively extends to the case of $m>n$ as follows.

Remark 3.8**.**

Fix $n\geq 4$ and let $P_{1},\dots,P_{n+k}\subset{\mathbb{R}}^{n}$ be $n$-dimensional lattice polytopes with $\operatorname{md}(P_{1},\dots,P_{n+k})=1$. Then, up to equivalence, the $(n+k)$-tuple $P_{1},\dots,P_{n+k}$ either belongs to a finite list of exceptions or there is a lattice projection $\varphi\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n-1}$ such that $\varphi(P_{i})=\Delta_{n-1}$ for all $1\leq i\leq n+k$.

One can see this with an induction argument on $k$, where the base case is given by Theorem 1.5. Indeed, let $P_{1},\dots,P_{n+k+1}\subset{\mathbb{R}}^{n}$ be an $(n+k+1)$-tuple of $n$-dimensional lattice polytopes with $\operatorname{md}(P_{1},\dots,P_{n+k+1})=1$. One easily verifies that this implies that any $(n+k)$-subtuple of $P_{1},\dots,P_{n+k+1}$ has mixed degree at most $1$. Analogously to the proof of Theorem 1.5 one can distinguish three cases depending on how many $(n+k)$-subtuples of $P_{1},\dots,P_{n+k+1}$ do not admit a common lattice projection onto translates of $\Delta_{n-1}$, and use the induction hypothesis.

4. The $3$-dimensional case

This section is devoted to the proof of Theorem 1.6, giving a classification of triples of $n$-dimensional lattice polytopes $P_{1},P_{2},P_{3}\subset{\mathbb{R}}^{3}$ of mixed degree one. Note that from the proof of Theorem 1.5 it follows that the number of such triples is finite, if we assume at least one of the subpairs of $P_{1},P_{2},P_{3}$ to be exceptional. Here we first classify, up to equivalence, these finitely many triples. In Proposition 4.7 we show that there are non-trivial infinite $1$-parameter families of triples.

As an intermediate step towards the classification of triples of lattice polytopes of mixed degree one with at least one exceptional subpair we calculate all (equivalence classes of) exceptional pairs of $3$-dimensional lattice polytopes. In order to do that we consider the list of maximal hollow $3$-dimensional lattice polytopes classified by Averkov–Wagner–Weismantel [AWW11] (see also [AKW17]), and compute all subpolytopes of the maximal hollow lattice polytopes that have lattice width greater than one.

Proposition 4.1** ([AWW11, Corollary 1]).**

Let $P\subset{\mathbb{R}}^{3}$ be a hollow $3$-dimensional lattice polytope of width at least two. Then, up to equivalence, $P$ is contained either in the unbounded polyhedron $2\Delta_{2}\times{\mathbb{R}}$ or in one of 12 maximal hollow lattice polytopes.

As we are interested in obtaining a list of exceptional pairs $P,Q\subset{\mathbb{R}}^{3}$ we use an implementation in Magma in order to compute the decompositions of all subpolytopes of the $12$ maximal hollow polytopes into Minkowski sums of two $3$-dimensional lattice polytopes. Afterwards we determine those pairs that actually do not admit a common projection onto translates of $\Delta_{2}$ and then determine equivalent pairs using Lemma 2.4.

Corollary 4.2**.**

There are, up to equivalence, 32 pairs of $3$-dimensional lattice polytopes whose Minkowski sum is hollow and that do not admit a common projection onto translates of $\Delta_{2}$.

We use this classification in order to compute all triples of lattice polytopes $P_{1},P_{2},P_{3}\subset{\mathbb{R}}^{3}$ of mixed degree one with at least two exceptional subpairs as follows.

Assume that $P_{1},P_{2}$ and $P_{1},P_{3}$ are exceptional pairs. Then there exist two pairs $A,B$ and $C,D$ out of the 32 of Corollary 4.2 such that $A,B$ is equivalent to $P_{1},P_{2}$ and $C,D$ is equivalent to $P_{1},P_{3}$. We can suppose that $P_{1}$ is equal to $A$ and equivalent to $C$. Thus there exists an affine lattice-preserving transformation $\varphi$ mapping $C$ to $A=P_{1}$ such that the triple $P_{1},P_{2},P_{3}$ is equivalent to the triple $A,B,\varphi(D)$.

This justifies the following algorithm to construct all the triples $P_{1},P_{2},P_{3}$ containing at least two exceptional subpairs: we iterate over all the pairs of ordered pairs $A,B$ and $C,D$ of Corollary 4.2, and, whenever there exists an affine lattice-preserving transformation $\varphi$ mapping $C$ to $A$, check if the triple $A,B,\varphi(\psi(D))$ has mixed degree one, where $\psi$ ranges among all the possible affine automorphisms of $C$ (and therefore $\varphi\circ\psi$ ranges among all affine lattice-preserving transformations sending $C$ to $A$). Equivalent triples can be removed using the criterion following from Lemma 2.4. An implementation in Magma yields the following result proving parts (i)-(ii) of Theorem 1.6.

Proposition 4.3**.**

There are, up to equivalence, 170 triples of $3$-dimensional lattice polytopes of mixed degree one having two or three exceptional subpairs. In the first case there are 29 triples, in the latter there are 141.

We now discuss the case of triples of lattice polytopes of mixed degree one having exactly one exceptional subpair. Specifically, $P_{1},P_{2},P_{3}\subset{\mathbb{R}}^{3}$ is a triple of $3$-dimensional lattice polytopes with $\operatorname{md}(P_{1},P_{2},P_{3})=1$ and (without loss of generality) there are two different lattice projections $\varphi_{2}\colon{\mathbb{R}}^{3}\to{\mathbb{R}}^{2}$ and $\varphi_{3}\colon{\mathbb{R}}^{3}\to{\mathbb{R}}^{2}$ where $\varphi_{k}$ maps $P_{i}$ and $P_{j}$ to translates of $\Delta_{2}$ whenever $i,j\neq k$. In particular $P_{1}$ is a lattice polytope with two different lattice projections onto $\Delta_{2}$ (and therefore by Lemma 3.2 is equivalent either to $\Delta_{3}$ or to $\mathcal{P}(\square_{2})$) and $P_{2},P_{3}$ is an exceptional pair. Note that $P_{1}$ must be contained in both the infinite prisms $C_{2}\mathrel{\mathop{:}}=P_{3}+\ker\varphi_{2}+u$ and $C_{3}\mathrel{\mathop{:}}=P_{2}+\ker\varphi_{3}+v$, for some translation vectors $u,v\in{\mathbb{Z}}^{3}$.

In order to classify all such triples we use the fact that we may choose $P_{2},P_{3}$ from the list of 32 exceptional pairs of Corollary 4.2. Given an exceptional pair $P_{2},P_{3}$, we iterate over all the possible pairs of lattice projections $\varphi_{3},\varphi_{2}$, such that $\varphi_{3}(P_{2})$ and $\varphi_{2}(P_{3})$ are unimodular triangles. Each such choice determines two infinite prisms $C_{3}\mathrel{\mathop{:}}=P_{2}+\ker\varphi_{3}$ and $C_{2}\mathrel{\mathop{:}}=P_{3}+\ker\varphi_{2}$. We know that any lattice polytope $P_{1}\subset{\mathbb{R}}^{3}$, such that $\varphi_{3}(P_{1})$ and $\varphi_{2}(P_{1})$ are translates of $\varphi_{3}(P_{2})$ and $\varphi_{2}(P_{3})$ respectively, is contained in both the infinite prisms $C_{2}\mathrel{\mathop{:}}=P_{3}+\ker\varphi_{2}+u$ and $C_{3}\mathrel{\mathop{:}}=P_{2}+\ker\varphi_{3}+v$, for some translation vectors $u,v\in{\mathbb{Z}}^{3}$. Up to translation of $P_{1}$ we may assume $u=\textnormal{{0}}$. By Lemma 3.3 it suffices to find one choice of $v\in{\mathbb{Z}}^{3}$ such that $C_{2}$ and $C_{3}$ intersect in a full-dimensional lattice polytope, in order to determine the inclusion-maximal choice for $P_{1}$ up to translation. Furthermore, there are only finitely many choices for $v\in{\mathbb{Z}}^{3}$ to check for the existence of a full-dimensional intersection of $C_{2}$ and $C_{3}$ as we may suppose $P_{2}$ and $P_{3}$ to have a common vertex. This is due to the fact that, if $C_{2}$ and $C_{3}$ intersect in a full-dimensional lattice polytope, then one may translate $P_{2}$ along $\ker\varphi_{3}$ and $P_{3}$ along $\ker\varphi_{2}$ without changing the infinite prisms. It therefore suffices to restrict to translation vectors $v$ that map a vertex of $P_{2}$ to a vertex of $P_{3}$. Thus we can determine, up to equivalence, all inclusion-maximal $P_{1}$ as above, form triples for all subpolytopes of $P_{1}$ and remove equivalent triples using Lemma 2.4. An implementation in Magma yields the following result proving part (iii) of Theorem 1.6.

Proposition 4.4**.**

There are, up to equivalence, 82 triples of $3$-dimensional lattice polytopes of mixed degree one having exactly one exceptional subpair.

In the remaining part of this section we are going to deal with non-trivial triples not having any exceptional subpair in order to prove part (iv) of Theorem 1.6.

Lemma 4.5**.**

Let $P_{1},P_{2},P_{3}\subset{\mathbb{R}}^{n}$ be lattice polytopes, and $\varphi_{1},\varphi_{2},\varphi_{3}\colon{\mathbb{R}}^{3}\to{\mathbb{R}}^{2}$ be lattice projections such that, for all $1\leq i,j,k\leq 3$, the images $\varphi_{k}(P_{i})$ and $\varphi_{k}(P_{j})$ are translates of $\Delta_{2}$ if and only if $i,j\neq k$. Let $v_{i}\in{\mathbb{Z}}^{3}$ be the projection direction of $\varphi_{i}$ for $1\leq i\leq 3$. Then $v_{i}$ and $v_{j}$ are part of a lattice basis of ${\mathbb{Z}}^{3}$, for all $1\leq i,j\leq 3$. Moreover, if $v_{1},v_{2},v_{3}$ linearly span ${\mathbb{R}}^{3}$, then they form a lattice basis of ${\mathbb{Z}}^{3}$.

Proof.

For $1\leq k\leq 3$ let $C_{k}$ be the infinite prism $\varphi_{k}(P_{i})+{\mathbb{R}}v_{k}$, for some $i\neq k$. Note that, up to translation, this does not depend on the choice of $i$ as both $P_{i}$ and $P_{j}$ are contained in different translates of $C_{k}$, whenever $i,j\neq k$. We now fix any of the infinite prisms, say $C_{1}$. For simplicity we suppose $C_{1}=(\{0\}\times\Delta_{2})+{\mathbb{R}}\textnormal{{e}}_{1}$ and $P_{2},P_{3}\subset C_{1}$. In this way we avoid dealing with translations. Note that $v_{2}$ is parallel to an edge of $P_{3}$, and $v_{3}$ is parallel to an edge of $P_{2}$. Since both edges are contained in $C_{1}$, they project along $\textnormal{{e}}_{1}$ either to the same side of the triangle $\varphi_{1}(P_{2})=\varphi_{1}(P_{3})=\Delta_{2}$, or to two adjacent sides. In the second case $\textnormal{{e}}_{1}$, $v_{2}$ and $v_{3}$ linearly span ${\mathbb{R}}^{3}$ and it is easy to verify that they form a lattice basis of ${\mathbb{Z}}^{3}$. In the first case $\textnormal{{e}}_{1}$, $v_{2}$ and $v_{3}$ span a plane, and from Lemma 3.2 it follows that any two of them are part of a lattice basis of ${\mathbb{Z}}^{3}$. ∎

Proposition 4.6**.**

There are, up to equivalence, 27 triples of $3$-dimensional lattice polytopes $P_{1},P_{2},P_{3}\subset{\mathbb{R}}^{3}$ satisfying the hypotheses of Lemma 4.5 for projection directions $v_{1},v_{2},v_{3}\in{\mathbb{Z}}^{3}$ that linearly span ${\mathbb{R}}^{3}$. All of them are, up to equivalence, contained in one of the following three inclusion-maximal triples of mixed degree one:

• •

the maximal triple given by the following three reflections of $\mathcal{P}(\square_{2})$

• $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{2},\textnormal{{e}}_{3},\textnormal{{e}}_{2}+\textnormal{{e}}_{3},\textnormal{{e}}_{1})$,
• $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{3},\textnormal{{e}}_{1}+\textnormal{{e}}_{3},\textnormal{{e}}_{2})$,
• $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},\textnormal{{e}}_{3})=\mathcal{P}(\square_{2})$,
• •

the maximal triple

• $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{3},\textnormal{{e}}_{1}+\textnormal{{e}}_{2})$,
• $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{3},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{3})$,
• $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},\textnormal{{e}}_{3})=\mathcal{P}(\square_{2})$.
• •

and the maximal triple

• $\operatorname{conv}(\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},\textnormal{{e}}_{2}+\textnormal{{e}}_{3})$,
• $\operatorname{conv}(\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{3})$,
• $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}+\textnormal{{e}}_{2},\textnormal{{e}}_{3})=\mathcal{P}(\square_{2})$.

Proof.

By Lemma 4.5 we may assume $v_{1},v_{2},v_{3}$ to be $\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{3}$ respectively, and that two primitive segments parallel to the directions $\textnormal{{e}}_{2}$ and $\textnormal{{e}}_{3}$ are contained in $C_{1}$. This restricts $C_{1}$ to be, up to translation, one of the four infinite prisms of the form $\operatorname{conv}(\textnormal{{0}},\pm\textnormal{{e}}_{2},\pm\textnormal{{e}}_{3})+{\mathbb{R}}\textnormal{{e}}_{1}$. In particular, up to translation, $C_{1}$ is contained in the infinite prism $\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{2})+\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{3})+{\mathbb{R}}\textnormal{{e}}_{1}$. Similarly, $C_{2}\subset\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1})+\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{3})+{\mathbb{R}}\textnormal{{e}}_{2}$ and $C_{3}\subset\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1})+\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{2})+{\mathbb{R}}\textnormal{{e}}_{3}$. In particular all the $P_{i}$ are, up to translations, subpolytopes of the unit cube $\square_{3}=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1})+\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{2})+\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{3})$, which leaves finitely many cases that we check computationally. ∎

From the proof of Proposition 4.6 it is clear that all the maximal triples from Proposition 4.6 are actually contained inside the triple consisting of three copies of the unit cube $\square_{3}$. Note however that one has $\operatorname{md}(\square_{3},\square_{3},\square_{3})>1$, as the Minkowski sum $\square_{3}+\square_{3}$ has an interior lattice point.

Proposition 4.7**.**

There are, up to equivalence, infinitely many triples of $3$-dimensional lattice polytopes $P_{1},P_{2},P_{3}\subset{\mathbb{R}}^{3}$ satisfying the hypothesis of Lemma 4.5 for projection directions $v_{1},v_{2},v_{3}\in{\mathbb{Z}}^{3}$ that linearly span {\mathbb{R}}^{2}\times\mathopen{}\mathclose{{}\left\{{0}}\right\}. All of them, up to equivalence, are contained in one of the following triples of mixed degree two given by the parallelepipeds $Q_{k},R_{k},\square_{3}$ for some $k\in{\mathbb{Z}}_{\geq 0}$, where

[TABLE]

They can be covered by a finite number of $1$-parameter families. In particular, if we denote by $\psi_{1}^{k}$ the shearing $(x,y,z)\mapsto(x,y+kz,z)$ and by $\psi_{2}^{k}$ the shearing $(x,y,z)\mapsto(x+kz,y,z)$, one may choose $1$-parameter families of the form \mathopen{}\mathclose{{}\left\{{\psi_{1}^{k}(P^{0}_{1}),\psi_{2}^{k}(P^{0}_{2}),P_{3}}}\right\}_{k\in{\mathbb{Z}}_{\geq 0}} for all subpolytopes $P_{1}^{0}\subset Q_{0},P_{2}^{0}\subset R_{0}$ and $P_{3}\subset\square_{3}$ satisfying $\operatorname{md}(P_{1}^{0},P_{2}^{0},P_{3})=1$.

Proof.

By Lemma 4.5 we may assume $v_{1},v_{2},v_{3}$ to be $\textnormal{{e}}_{1},\textnormal{{e}}_{2},\textnormal{{e}}_{1}-\textnormal{{e}}_{2}$. Here, the assumption $v_{3}=\textnormal{{e}}_{1}-\textnormal{{e}}_{2}$ follows from the fact that both the pairs $\textnormal{{e}}_{1},v_{3}$ and $\textnormal{{e}}_{2},v_{3}$ need to be part of a lattice basis of ${\mathbb{Z}}^{3}$, and the projection directions $v_{i}$ may be chosen with arbitrary sign. By Lemma 3.2 the polytope $P_{3}$, which projects onto $\Delta_{2}$ along the directions $\textnormal{{e}}_{1}$ and $\textnormal{{e}}_{2}$, can be fixed to be in the unit cube $\square_{3}$. Consequently we can assume $C_{1},C_{2}$ to be in the infinite prisms $\square_{3}+{\mathbb{R}}\textnormal{{e}}_{1}$ and $\square_{3}+{\mathbb{R}}\textnormal{{e}}_{2}$, respectively. Finally, we assume $P_{1}$ and $P_{2}$ to be in the infinite prisms $C_{2}$ and $C_{1}$, respectively. Now consider the linear functional $f$ defined by $(x,y,z)\mapsto x+y$. Consider a lattice point $v_{0}\in P_{1}\cap({\mathbb{R}}^{2}\times\{0\})$ minimizing $f$. Since $P_{1}$ projects onto $\Delta_{2}$ along the direction $\textnormal{{e}}_{1}-\textnormal{{e}}_{2}$, one verifies that for any other point $u_{0}\in P_{1}\cap({\mathbb{R}}^{2}\times\{0\})$ one has $f(v_{0})\leq f(u_{0})\leq f(v_{0})+1$. Analogously, if $v_{1}$ is a lattice point in $P_{1}\cap({\mathbb{R}}^{2}\times\{1\})$ minimizing $f$, then $f(v_{1})\leq f(u_{1})\leq f(v_{1})+1$ for all $u_{1}\in P_{1}\cap({\mathbb{R}}^{2}\times\{1\})$. Since we are free to translate $P_{1}$ along $\textnormal{{e}}_{2}$, we can suppose $f(v_{0})=0$ and we denote $k=f(v_{1})$. As a consequence, $P_{1}\cap({\mathbb{R}}^{2}\times\{0\})$ is contained in the parallelogram $q_{0}\mathrel{\mathop{:}}=\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{1}-\textnormal{{e}}_{2})+\operatorname{conv}(\textnormal{{0}},\textnormal{{e}}_{2})$. Analogously $P_{1}\cap({\mathbb{R}}^{2}\times\{1\})$ is contained in the parallelogram $q_{1}\mathrel{\mathop{:}}=q_{0}+k\textnormal{{e}}_{2}+\textnormal{{e}}_{3}$. In particular $P_{1}$ is contained in the parallelepiped $\operatorname{conv}(q_{0}\cup q_{1})=Q_{k}$. Therefore $C_{3}$ is contained in the infinite prism $Q_{k}+{\mathbb{R}}(\textnormal{{e}}_{1}-\textnormal{{e}}_{2})$. This completely determines the parallelepiped $R_{k}=C_{1}\cap C_{3}$, satisfying $R_{k}\supset P_{2}$. It is easy to verify that the triple $Q_{k},R_{k},\square_{3}$ is equivalent to the triple $Q_{-k},R_{-k},\square_{3}$, so one can always assume $k\in{\mathbb{Z}}_{\geq 0}$.

In order to see that the set of triples that are subtriples of $Q_{k},R_{k},\square_{3}$ for some $k\in{\mathbb{Z}}_{\geq 0}$ can be covered by $1$-parameter families as claimed it suffices to notice that any subtriple of $Q_{k},R_{k},\square_{3}$ can be written as $\psi_{1}^{k}(P_{1}^{0}),\psi_{2}^{k}(P_{2}^{0}),\square_{3}$ for subpolytopes $P_{1}^{0}\subset Q_{0},P_{2}^{0}\subset R_{0}$ and $P_{3}\subset\square_{3}$. The fact that any family \mathopen{}\mathclose{{}\left\{{\psi_{1}^{k}(P^{0}_{1}),\psi_{2}^{k}(P^{0}_{2}),P_{3}}}\right\}_{k\in{\mathbb{Z}}_{\geq 0}} for subpolytopes $P_{1}^{0}\subset Q_{0},P_{2}^{0}\subset R_{0}$ and $P_{3}\subset\square_{3}$ actually contains infinitely many non-equivalent triples can be verified by picking edges $E_{i}\subset P_{i}$ for $1\leq i\leq 3$ between vertices on height 0 and 1, and noticing that the volume of the parallelepiped $E_{1}+E_{2}+E_{3}$ grows quadratically in $k$. An example of one of these inifinite $1$-parameter families is given in Example 1.7. ∎

A computer assisted search for mixed degree one triples in $Q_{k},R_{k},\square_{3}$ for small values of $k$ shows that there are 51 non-equivalent triples when $k=0$, and $36$ for larger values of $k$, where, for each $k$, the overlaps that occur for preceding values of $k$ are excluded.

Let us finally mention that the 252 triples classified in Theorem 1.6 (i) –(iii) can all be found as subtriples of six special triples which are maximal with respect to inclusion. We have verified this computationally by enumerating subtriples of the six special ones.

Corollary 4.8**.**

All triples $P_{1},P_{2},P_{3}\subset{\mathbb{R}}^{3}$ of $3$-dimensional lattice polytopes of mixed degree one of types (i) –(iii) from Theorem 1.6 are, up to equivalence, contained in one of the following $6$ maximal triples:

1. (a)

the maximal triple ${\mathcal{P}}(2\Delta_{2}),{\mathcal{P}}(2\Delta_{2}),{\mathcal{P}}(2\Delta_{2})$, 2. (b)

the maximal triple $2\Delta_{3},\Delta_{3},\Delta_{3}$, 3. (c)

the maximal triple $\{\operatorname{conv}(\textnormal{{0}},2\textnormal{{e}}_{i},\textnormal{{e}}_{j},\textnormal{{e}}_{k})\colon i,j,k\in[3]\text{ pairwise different}\}$, 4. (d)

the maximal triple

[TABLE] 5. (e)

the maximal triple

[TABLE] 6. (f)

the maximal triple

[TABLE]

Note that the maximal triples (a) and (b) of Corollary 4.8 admit direct generalizations to an arbitrary dimension $n$ that are of mixed degree one. Furthermore it follows from the proof of Theorem 1.5 that there are no $n$-tuples of a type analogous to type (iv) of Theorem 1.6 for $n\geq 4$. A bold guess for an answer to Question 3.7 would be that for arbitrary $n\geq 4$ (or $n$ large enough) all exceptions of $n$-tuples of mixed degree one are contained in one of these generalizations, that is either in ${\mathcal{P}}^{n-2}(2\Delta_{2}),\dots,{\mathcal{P}}^{n-2}(2\Delta_{2})$ or $2\Delta_{n},\Delta_{n},\dots,\Delta_{n}$ (as it can be easily verified that the straightforward generalization of the maximal family (c) to dimension $n\geq 4$ does not yield $n$-tuples of mixed degree one).