Inverse Blaschke-Santal\'o inequality for convex curves enclosing the origin several times

TL;DR

This paper investigates a conjecture on the minimal volume product of convex curves enclosing the origin multiple times, providing proofs for specific cases and bounds for others, advancing understanding of geometric inequalities.

Contribution

The paper proves the conjecture for certain convex polygons with equal central angles and establishes bounds for other cases, extending previous results on volume product inequalities.

Findings

Proved the conjecture for convex n-gons with 2k+1 ≤ n ≤ 4k and equal central angles.

Established that for n ≥ 4k+1, stationary volume product values occur when vertices lie on the unit circle with equal angles.

Provided a bound of approximately 0.43 for the conjecture in inscribed polygons with 2k+1 ≤ n.

Abstract

H. Guggenheimer generalized the planar volume product problem for locally convex curves $C$ enclosing the origin $k \ge 2$ times. He conjectured that the minimal volume product $V(C)V(C^*)$ for these curves is attained if the curve consists of the longest diagonals of a regular $(2k+1)$-gon, with centre $0$, these diagonals taken always in the positive orientation. This conjectured minimum is of the form $k^2 + O(k)$. We investigate special cases of this conjecture. We prove it for locally convex $n$-gons with $2k+1 \le n \le 4k$, if the central angles at $0$ of all sides are equal to $2k \pi /n$. For $4k+1 \le n$ we prove that for locally convex $n$-gons enclosing the origin $k \ge 2$ times the critical (stationary) values of the volume product $V(K)V(K^*)$ are attained exactly when up to a non-singular linear map the vertices lie on the unit circle about $0$, and the central angles of all sides are equal to $2k \pi /n$. For locally convex $n$-gons enclosing the origin $k \ge 2$ times, and inscribed to the unit circle, with $2k+1 \le n$, we prove the conjecture up to a multiplicative factor about $0.43$.

Equations

$$c_{1,d}=\min \{V(K)V(K^{\ast })\mid K\subset {\mathbb{R}}^d\text{ is a convex body with }0\in \text{int}K\}.$$

$$\{\begin{array}{l}{c}_{2,d}^{\prime }=\max \{V(K)\cdot \min \{V\left((K-x{)}^{\ast })\right)\mid x\in \text{int}K\}\mid K\subset {\mathbb{R}}^d\text{ is a convex body}\}\\ =\max \{V(K)V\left({\left(K-s(K)\right)}^{\ast }\right)\mid K\subset {\mathbb{R}}^d\text{ is a convex body}\}.\end{array}$$

$${\int }_0^{2k\pi }\varrho (\phi {)}^{2}d\phi /2$$

$$\{\begin{array}{l}\text{the value of }V[{\left(C-x\right)}^{\ast }]\text{ tends to infinity uniformly if }\\ x\text{ approaches the closest point of }C\text{ on any ray from }0.\end{array}$$

$$V\left(i(M)\right)=V(i,M)={\int }_M\Vert i(m)\Vert ⟨i(m),n\left(i(m)\right)dS\left(i(m)\right)/d,$$

$$V(C)\ge V(E_2)=\pi /\epsilon ,\text{ and }V\left((C-x{)}^{\ast }\right)\ge V\left((E_1-x{)}^{\ast }\right)\ge V(E_1^{\ast })=\pi /\epsilon ,$$

$$\{\begin{array}{l}V(C)V[{\left(C-s(C)\right)}^{\ast }]=\\ V(C)\inf \{V\left((C-x{)}^{\ast }\right)\mid x\in \text{kernel of }i(M)\}\ge {\pi }^2/{\epsilon }^2.\end{array}$$

$$\{\begin{array}{l}V(C)V[{\left(C-s(C)\right)}^{\ast }]=\\ V(C)\inf \{V\left((C-x{)}^{\ast }\right)\mid x\in \text{kernel of }i(M)\}\ge {\kappa }_d^2/{\epsilon }^d.\end{array}$$

$$\{\begin{array}{l}V(iM)V\left((iM{)}^{\ast }\right)\ge {\left({\prod }_{j=1}^{l}V(i{M}_j)\right)}^{1/l}\cdot \\ {\left({\prod }_{j=1}^{l}V\left((i{M}_j{)}^{\ast }\right)\right)}^{1/l}={\left({\prod }_{j=1}^{l}V(i{M}_j)V\left((i{M}_j{)}^{\ast }\right)\right)}^{1/l}.\end{array}$$

$$V(C)=\sum _{i=1}^n{\varrho }_i{\varrho }_{i+1}(\sin {\vartheta }_i)/2.$$

$$\{\begin{array}{l}4V(Q_i)=[1/{\varrho }_i^2+1/{\varrho }_{i+1}^2]\cdot [-{\cos }^2(\vartheta /2)\cot (\vartheta /2)+{\sin }^2(\vartheta /2)\tan (\vartheta /2)]\\ +[2\cdot (1/{\varrho }_i)\cdot (1/{\varrho }_{i+1})]\cdot [{\cos }^2(\vartheta /2)\cot (\vartheta /2)+{\sin }^2(\vartheta /2)\tan (\vartheta /2)+\sin \vartheta ].\end{array}$$

$$\{\begin{array}{l}[{\cos }^2(\vartheta /2)\cot (\vartheta /2)]/[{\sin }^2(\vartheta /2)\tan (\vartheta /2)]={\cot }^4(\vartheta /2)=\\ {\cot }^4\left((2k\pi /n)/2\right)\le {\cot }^4\left(\left(2k\pi /(4k)\right)/2\right)={\cot }^4(\pi /4)=1,\end{array}$$

$$\{\begin{array}{l}V(C)V(C^{\ast })=[{\sum }_{i=1}^n{\varrho }_i{\varrho }_{i+1}(\sin \vartheta )/2]\cdot \\ {\sum }_{i=1}^n\left[(1/{\varrho }_i^2+1/{\varrho }_{i+1}^2)\cdot F(\vartheta )/4+\left(2/({\varrho }_i{\varrho }_{i+1})\right)\cdot G(\vartheta )/4\right].\end{array}$$

$$\{\begin{array}{l}V(C)V(C^{\ast })\ge n\cdot [{\prod }_{i=1}^n({\varrho }_i{\varrho }_{i+1}){]}^{1/n}[(\sin \vartheta )/2]\cdot \\ n\cdot {\prod }_{i=1}^n{\left[(1/{\varrho }_i^2+1/{\varrho }_{i+1}^2)\cdot F(\vartheta )/4+\left(2/({\varrho }_i{\varrho }_{i+1})\right)\cdot G(\vartheta )/4\right]}^{1/n}=\\ n^2\cdot [(\sin \vartheta )/2]\cdot {\prod }_{i=1}^n{\left[({\varrho }_{i+1}/{\varrho }_i+{\varrho }_i/{\varrho }_{i+1})\cdot F(\vartheta )/4+2\cdot G(\vartheta )/4\right]}^{1/n}.\end{array}$$

$${\varrho }_i={\varrho }_{i+2}$$

$$({\varrho }_{i+1}/{\varrho }_i+{\varrho }_i/{\varrho }_{i+1})\cdot F(\vartheta )/4\ge 2\cdot F(\vartheta )/4,$$

$$\{\begin{array}{l}\text{either }F(\vartheta )=0,\text{ i.e., }n=4k,\text{ or }F(\vartheta )>0,\text{ i.e.,}\\ 2k+1\le n\le 4k-1,\text{ and }{\varrho }_i={\varrho }_{i+1}\text{ for each }i.\end{array}$$

$$\{\begin{array}{l}V(C)V(C^{\ast })\ge n^2\cdot [(\sin \vartheta )/2]\cdot {\prod }_{i=1}^n(2\cdot F(\vartheta )/4+2\cdot G(\vartheta )/4{)}^{1/n}=\\ n^2\cdot [(\sin \vartheta )/2]\cdot \left(F(\vartheta )+G(\vartheta )\right)/2=\\ n^2\cdot [(\sin \vartheta )/2]\cdot \sin (\vartheta /2)\left(\cos (\vartheta /2)+\sin (\vartheta /2)\tan (\vartheta /2)\right)=\\ n^2{\sin }^2(\vartheta /2)=n^2{\sin }^2(k\pi /n),\end{array}$$

$$-\sin t/(1-\cos t{)}^2=-\cos (t/2)/[2{\sin }^3(t/2)],$$

$$(t+t{\cos }^{2}t-2\cos t\sin t)/{\sin }^{3}t,$$

$$t>2\cos t\sin t/(1+{\cos }^{2}t),\text{ i.e., }s\ge (4\sin s)/(3+\cos s),$$

$$1>(12\cos s+4)/(3+\cos s{)}^2.$$

$$(5-\cos t)(1-\cos t)>0,$$

$$\sum _{i=1}^n{\vartheta }_i=2k\pi .$$

$$V(C)=\sum _{i=1}^n(\sin {\vartheta }_i)/2,\text{ and }V(C^{\ast })=\sum _{i=1}^n\tan ({\vartheta }_i/2).$$

$$\{\begin{array}{l}V(C)C(C^{\ast })\ge \sum \{(\sin {\vartheta }_i)/2\mid {\vartheta }_i\in (0,c]\}\cdot \sum \{\tan ({\vartheta }_i/2)\mid {\vartheta }_i\in (0,c]\}+\\ \sum \{(\sin {\vartheta }_i)/2\mid {\vartheta }_i\in (c,\pi )\}\cdot \sum \{\tan ({\vartheta }_i/2)\mid {\vartheta }_i\in (c,\pi )\}.\end{array}$$

$$k_{1}:=\sum\{\vartheta_{i}\mid\vartheta_{i}\in(0,c]\big{/}\pi{\text{ and }}k_{2}:=\sum\{\vartheta_{i}\mid\vartheta_{i}\in(c,\pi)\big{/}\pi,$$

$$k_1,k_2\in [0,2k],\text{ and }{k}_1+k_2=2k.$$

$$(\sin {\vartheta }_i)/2\ge ({\vartheta }_i/2)\cdot (\sin c)/c\text{ and }\tan ({\vartheta }_i/2)\ge {\vartheta }_i/2.$$

$$\{\begin{array}{l}(\sin {\vartheta }_i)/2=\left(\sin (\pi -{\vartheta }_i)\right)/2\ge [(\pi -{\vartheta }_i)/2]\cdot [\left(\sin (\pi -c)\right)/(\pi -c)]\\ \text{and }\tan ({\vartheta }_i/2)=1/\tan \left((\pi -{\vartheta }_i)/2\right)\ge \\ [1/\left((\pi -{\vartheta }_i)/2\right)]\cdot [\left(\pi -c)/2\right)/\tan \left((\pi -c)/2\right)].\end{array}$$

Full text

Inverse Blaschke-Santaló inequality for convex curves enclosing the

origin several times

K. J. Böröczky, E. Makai, Jr.

Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences

H-1364 Budapest, Pf. 127, Hungary

http://www.renyi.mta.hu/~carlos, http://www.renyi.mta.hu/~makai

E-mail: carlosrenyi.mta.hu, makai.endrerenyi.mta.hu

H. Guggenheimer generalized the planar volume product problem for locally convex curves $C$ enclosing the origin $k\geq 2$ times. He conjectured that the minimal volume product $V(C)V(C^{*})$ for these curves is attained if the curve consists of the longest diagonals of a regular $(2k+1)$-gon with centre [math], taken always in the positive orientation. This conjectured minimum is of the form $k^{2}+O(k)$. We investigate special cases of this conjecture. We prove it for locally convex $n$-gons with $2k+1\leq n\leq 4k$, if the central angles at [math] of all sides are equal to $2k\pi/n$. For $4k+1\leq n$ we prove that for locally convex $n$-gons enclosing the origin $k\geq 2$ times the critical (stationary) values of the volume product $V(K)V(K^{*})$ are attained exactly when up to a non-singular linear map the vertices lie on the unit circle about [math], and the central angles of all sides are equal to $2k\pi/n$. For locally convex $n$-gons enclosing the origin $k\geq 2$ times, and inscribed to the unit circle, with $2k+1\leq n$, we prove the conjecture up to a multiplicative factor about $0.43$.

2010 Mathematics Subject Classification. Primary: 52A40. Secondary: 52A30, 52A10

Key words and phrases. inverse Blaschke-Santaló inequality, convex curves enclosing the origin several times

1. Introduction

We begin with some notations, and some well-known facts about the volume product. Cf., e.g., [L], [BMMS] and [Mak].

A convex body  in ${\mathbb{R}}^{d}$ is a compact convex set with nonempty interior. For $K\subset{\mathbb{R}}^{d}$ a convex body with $0\in{\text{int}}\,K$ we write $K^{*}$ for its polar body, which also is a convex body, containing [math] in its interior. One is interested in the infimum and supremum of $V(K^{*})$, if $V(K)$ is given. These are of the form $c_{1,d}/V(K)$ and $c_{2,d}/V(K)$, for

certain constants $c_{i,d}$. (An important special case is that of [math]-symmetric convex bodies, which however we will not treat.)

We have

[TABLE]

K. Mahler [Mah39] conjectured that $c_{1,d}=(d+1)^{d+1}/(d!)^{2}$, with equality only for a simplex of barycentre [math], which is still unproved, although in many special cases it is known to hold. In particular, for $d=2$ this was proved by [Mah38], and the only case of equality is a triangle with barycentre at [math], as proved by [Me]. The conjecture is proved, up to a factor $\left(\pi/(2e)+o(1)\right)^{d}$, by [K]. (For the [math]-symmetric case the analogous minimum is conjectured to be $4^{d}/d!$, and is conjectured to be attained e.g. for a parallelotope, or a cross-polytope, and more generally, it is conjectured to be attained exactly for those bodies $K$, which, as unit balls of finite dimensional Banach spaces — i.e., of Minkowski spaces — can be obtained from $[-1,1]\subset{\mathbb{R}}$, by taking, in an arbitrary order, $l^{1}$-sums and $l^{\infty}$-sums of lower dimensional such Banach spaces. This conjecture is proved, up to a factor $\left(\pi/4+o(1)\right)^{d}$, by [K]. Quite recently the proof of the three-dimensional case of this conjecture, together with the conjectured equality cases, i.e., parallelepiped and affine regular octahedron, was announced in [IS], and was reassured in [I].)

However, $c_{2,d}=\infty$. Therefore one has to consider the minimax problem, i.e., the supremum of $\min\{V\left((K-x)^{*}\right)\mid x\in{\text{int}}\,K\}$. (The function $V\left((K-x)^{*}\right)$ is strictly convex for $x\in{\text{int}}\,K$ and tends to infinity if ${\text{dist}}\,(x,{\text{bd}}\,K)\to 0$, therefore this function has a unique minimum place, the so called Santaló point $s(K)$ of  $K$.)

This supremum is of the form $c^{\prime}_{2,d}/V(K)$, for a certain constant $c^{\prime}_{2,d}$. One has

[TABLE]

The theorem that $c^{\prime}_{2,d}=\kappa_{d}^{2}$ (where $\kappa_{d}$ is the volume of the unit ball of ${\mathbb{R}}^{d}$) was proved by W. Blaschke and L. Santaló, cf. [B] and [S]. The only case of equality is for the ellipsoid, which was proved by [SR], [P] and [MP].

We remark that $V(K)V(K^{*})$ is invariant under non-singular linear mappings, and $V(K)V\left(K-s(K)\right)$ is affine invariant. In particular, if $K$ admits an affinity with a single fixed point, then this point is the Santaló point of $K$.

These two quantities turn out to be in the cross-road of many disciplines: they arose in affine differential geometry (in [B]) and geometry of numbers (in [M39]), but also the [math]-symmetric case is very important in finite dimensional Banach spaces. Further such disciplines are discrete geometry, geometrical probabilities, integral geometry in Minkowski spaces, differential equations, and even theory of functions

of several complex variables.

Inverse Blaschke-Santaló inequality for convex curves enclosing the

origin several times

H. Guggenheimer [G] posed an interesting generalization of the planar volume product problem.

Definition Definition 1

(H. Guggenheimer, [G]) Let $k\geq 2$ be an integer. [G] defines the class ${\Cal{C}}_{k}^{2}$ of closed curves $C\subset{\mathbb{R}}^{2}\setminus\{0\}$ as follows. $C\in{\Cal{C}}_{k}^{2}$ is given in polar coordinates as the graph of its radial function $\varrho(\cdot)$, where $\varrho:[0,2k\pi]\to(0,\infty)$ (or one can say, $\varrho(\cdot)$ is defined on ${\mathbb{R}}$ and is $2k\pi$-periodic). (Therefore $C$ encircles the origin $k$ times in the positive sense — in other words, the winding number, i.e., index of $C$ with respect to [math] equals $k$.) Moreover, $C$ is at each of its points locally convex, that is, has local supporting lines at each of its points, and is seen from the origin always in the concave side. We topologize ${\Cal{C}}_{k}^{2}$ by the supremum distance of the radial functions, on $[0,2k\pi]$ (or on ${\mathbb{R}}$).

Such a curve also has a a support function $h:[0,2k\pi]\to(0,\infty)$ (or one can say, $h(\cdot)$ is defined on ${\mathbb{R}}$ and is $2k\pi$-periodic).

One can topologize ${\Cal{C}}_{k}^{2}$ also by the maximum distance of the support functions. The two definitions are equivalent: a basic neighbourhood of $C\in{\Cal{C}}_{k}^{2}$, with radial function $\varrho_{C}(\varphi)$, consists in any of the two cases of those curves $C^{\prime}\in{\Cal{C}}_{k}^{2}$, whose points $(\varphi,\varrho^{\prime})$ satisfy $(1-\varepsilon)\varrho_{C}(\varphi)\leq\varrho^{\prime}\leq(1+\varepsilon)\varrho_{C}(\varphi)$, for all $\varphi\in[0,2k\pi]$ (or the analogous inequality for the support functions), where $\varepsilon>0$ is arbitrary.

The area $V(C)$ enclosed by $C$ can be defined as

[TABLE]

(this is also the integral of the index of $C$ on the whole plane). One can define polarity on ${\Cal{C}}_{k}^{2}$ as usual (the polar curve is denoted by $C^{*}$, and has the same properties as $C$), and also the Santaló point as usual, with the usual characteristics. We will use the Santaló point only in the plane.

Observe that

[TABLE]

If the ray points to an angle $\varphi\in[0,2\pi)$, then the distance of this closest point is $\min\{\varrho(\varphi+2i\pi)\mid 0\leq i\leq k-1\}$. These closest points enclose a convex disc

containing [math] in its interior, we call it the kernel of $C$. In particular, the Santaló point belongs to the kernel of $C$.

The question is again, as in the usual case: given $V(C)$, what is the range of values of $V(C^{*})$? Again we have the usual affine invariance property, hence this question is again equivalent with the question of supremum/infimum, or possibly maximum/minimum of the product $V(C)V[\left(C-s(C)\right)^{*}]$. (Actually, for this only dilations with positive ratio would be sufficient.)

[G] also considers the special case of [math]-symmetric curves $C$, however does not clarify, what does he mean by this. We think that the natural way is the following: the radial (or support) function is not just $2k\pi$-periodic, but actually is $k\pi$-periodic. For $k$ even, this means just a curve of index $k/2$ about [math], traversed twice, but then the curve $C$ as a set is not [math]-symmetric in general. Its volume product is $4$ times the volume product of this curve of index $k/2$ about [math], hence this case is covered by the study of curves of index $k/2$ about [math], hence is not to be investigated. For $k$ odd, the curve $C$ as a set, is [math]-symmetric. [G] does not seem to recognize these two cases.

For dimension $d\geq 3$ there is an analogous definition, cf. Definition 2 below. For smooth manifolds $M,N$, an immersion $i:M\to N$ is a map everywhere of rank ${\text{dim}}\,M$. For topological manifolds $M,N$ we say that a map $i:M\to N$ is an immersion if $i$ is locally a homeomorphism onto its image. That is, each $m\in M$ has a neighbourhood $U$ in $M$ such that $i$, considered as a map $U\to i(U)$, is a homeomorphism.

Definition Definition 2

Let $d\geq 3$ and $k\geq 2$ be integers. We write ${\Cal{C}}^{d}_{k}$ for the set of immersed manifolds in ${\mathbb{R}}^{d}\setminus\{0\}$, by an immersion $i:M\to{\mathbb{R}}^{d}\setminus\{0\}$, where $M$ is a connected compact topological $(d-1)$-manifold, for which for any of point $m\in M$ we have that some open neighbourhood $U\subset M$ of $m\in M$ has an image $i(U)$ such that besides $i$ being a homeomorphism $U\to i(U)$, additionally we have that $i(U)$ is a relatively open subset of the boundary of some convex body $K_{m}$ with $0\in{\text{int}}\,K_{m}$.

In Definition 2 $i$ is not an embedding: actually, the restriction of $i$ to each linear $2$-subspace of ${\mathbb{R}}^{d}$ is not an embedding. The kernel of $i(M)$ is the maximal open star domain in ${\mathbb{R}}^{d}\setminus i(M)$. The enclosed volume is defined as usual, by

[TABLE]

where $\|\cdot\|$ is the norm, $dS\left(i(m)\right)$ is the surface area measure ($(d-1)$-Hausdorff measure) element on $i(M)$ at $i(m)$, and $n\left(i(m)\right)$ is the outer unit normal on $i(M)$ at $i(m)$, uniquely defined $dS\left(i(m)\right)$-almost everywhere. We will use the notation $V\left(i(M)\right)$, if the immersion $i$ is understood. Observe that necessariy $M$ is orientable. In fact, [math] lies always in one of the open halfspaces bounded by local supporting hyperplanes of $i(M)$. The outward normals of these local supporting hyperplanes of $i(M)$ will be considered as outward normals of the entire $i(M)$. (These considerations can be taken over also for $M$ disconnected.)

Below, in the proof of Proposition 3, we will see examples of such immersed manifolds. (They will come from mappings $S^{d-1}\to S^{d-1}$ of index $k$.) The significance of connectedness of $M$ will be explained later, in Remark 8.

For the planar case, [G] tries to apply some local arguments for the lower estimate. However, in absence of compactness, local arguments are definitely insufficient for this. And in fact, for each $k\geq 2$, the equivalence classes of the curves in ${\Cal{C}}^{2}_{k}$ with respect to nonsingular linear maps do not form a compact set in their natural topology, contrary to what is asserted in [G]. More exactly, we have Corollary 4 below.

Proposition 3

Let $d,k\geq 2$ be integers. Then the continuous affine invariant functional $V(C)V[\left(C-s(C)\right)^{*}]$ is unbounded above. In other words, there is no Blaschke-Santaló theorem for ${\Cal{C}}_{k}^{d}$.

Demonstration Proof

First let $d=2$. We write, as usual, $e_{1}=(1,0)$ and $e_{2}=(0,1)$. Let $\varepsilon>0$ be an arbitrarily small number. We define $C$ as follows. Let $E_{1}$ and $E_{2}$ be ellipses with equations $x^{2}+(y/\varepsilon)^{2}=1$ and $x^{2}+(y\varepsilon)^{2}=1$. We define $C$ as follows. First we traverse ${\text{bd}}\,E_{1}$ once, from $e_{1}$ to $e_{1}$, in the positive sense, and then in continuation we traverse ${\text{bd}}\,E_{2}$, $k-1$ times, from $e_{1}$ to $e_{1}$, in the positive sense. Let $x\in{\text{int}}\,(E_{1}\cap E_{2})={\text{int}}\,E_{1}$ (this set is the kernel of $C$). Then

[TABLE]

hence

[TABLE]

For $d=3$ we rotate the $2$-dimensional example about the $x_{1}$-axis, and in the same way we obtain the $(d+1)$-dimensional example from the $d$-dimensional example. Then we obtain, writing $\kappa_{d}$ for the volume of the unit ball in ${\mathbb{R}}^{d}$, that

[TABLE]

Corollary 4

Let $d=2$, and let $k\geq 2$ be an integer. Then the equivalence classes of the immersed manifolds $C\in{\Cal{C}}_{k}^{2}$ with respect to non-singular linear maps do not form a compact set in the quotient topology.

Demonstration Proof

By * the product $V(C)V\left(C-s(C)\right)=V(C)\inf\{V\left((C-x)^{*}\right)\mid x\in{\text{kernel of }}i(M)\}$ is continuous, is invariant with respect to non-singular linear maps, and is also unbounded above, by Proposition 3. Therefore the equivalence classes of the immersed manifolds from ${\Cal{C}}^{2}_{k}$ with respect to non-singular linear maps do not form a compact set in their quotient topology. $\blacksquare$

Thus the remaining question is whether there is an inverse Blaschke-Santaló inequality here. Below we will describe the (conjecturable) statement of [G], for the planar case. We begin with a notation.

Definition Notation 5

Let $k\geq 2$ and $n\geq 2k+1\geq 5$. Then $C_{n,k}$ denotes the following closed polygonal line. We consider a regular $n$-gon of centre [math], inscribed in the unit circle $S^{1}$ of centre [math], and we pass successively on its $k$-th smallest diagonals, always in the positive sense, until the sum of the central angles of the sides attains $2k\pi$. We write $(k,n)$ for the greatest common divisor of $k,n$. Then $C_{n,k}$ passes only on each $(k,n)$-th vertex of the regular $n$-gon, but passes through each of them $(k,n)$ times.

Definition Conjecture 6

(H. Guggenheimer [G], stated there as theorems)

1. For the general (i.e., not $k\pi$-periodic) case [G] considers $C_{2k+1,k}$. This has index $k$ with respect to [math], and is conjectured to give the minimal volume product $V(C)V(C^{*})$ for all $C\in{\Cal{C}}_{k}^{2}$.

2. For the $k\pi$-periodic case [G] considers $C_{2k+2,k}$. This has index $k$ with respect to [math], and is conjectured to give the minimal volume product $V(C)V(C^{*})$ for the $k\pi$-periodic case. (We have to remark that for $k$ even we have here a doubly traversed curve $C_{0}\in{\Cal{C}}_{k/2}$, and then $V(C)V(C^{*})=4V(C_{0})V(C_{0}^{*})$. Therefore part 1) of this conjecture, for index $k/2$, implies part 2) of this conjecture for index $k$. However, for $k$ odd, the conjectured $C_{2k+2,k}$ passes through all vertices of the regular $(2k+2)$-gon, through each of them just once, and then part 2) of this conjecture does not follow from its part 1). Moreover, for $k$ odd, the curve $C_{2k+2,k}$ is [math]-symmetric.)

Definition Remark 7

For $C=C_{n,k}$, with $n\geq 2k+1$, we have, writing $\vartheta_{i}$ for the central angle of the $i$’th side, that, $\sum_{i=1}^{n}\vartheta_{i}=2k\pi$, all $\vartheta_{i}$’s equal $2k\pi/n$, and V(C)V(C^{*})=[\sum_{i=1}^{n}(\sin\vartheta_{i})/2]\cdot[\sum_{i=1}^{n}\tan(\vartheta_{i}/2)]=n^{2}\sin^{2}(k\pi/n)=\big{[}[\left(\sin(k\pi/n)\right)/(k\pi/n)]\cdot k\pi\big{]}^{2}, which strictly increases with $n$. So for a minimum we must have the minimal possible $n$, i.e., $n=2k+1$ for the general case, and $n=2k+2$ for the $k\pi$-periodic case with $k$ odd. This is a small support for our conjecture. The respective values of $V(C)V(C^{*})$ in these two cases are $(2k+1)^{2}\sin^{2}\left(k\pi/(2k+1)\right)$, and $(2k+2)^{2}\sin^{2}\left(k\pi/(2k+2)\right)$, both of the form $4k^{2}+O(k)$.

Definition Remark 8

Suppose that $M$ has several connected components $M_{1},\ldots,M_{l}$ (finitely many by compactness of $M$), with respective indices, i.e., winding numbers, $k_{1},\ldots,k_{l}$, satisfying $0<k_{j}$ and $\sum_{j=1}^{l}k_{j}=k$. Then the inverse Blaschke-Santaló inequality can be asked also for $M$. However, this question can be reduced to the connected case, with smaller indices $k$. In fact, we have $V(iM)=\sum_{j=1}^{l}V(iM_{j})$ and similarly $V\left((iM)^{*}\right)=\sum_{j=1}^{l}V\left((iM_{j})^{*}\right)$, hence by the arithmetic-geometrical mean inequality

[TABLE]

Hence if we have some non-trivial lower estimates for the indices $k_{1},\ldots,k_{l}$, then this implies some nontrivial lower estimate for $V(iM)V\left((iM)^{*}\right)$. For the planar case, assuming that Conjecture 6 were valid for indices smaller than $k$, we would have Conjecture 6 for disconnected $M$ with index $k$.

We turn to the case $d=2$. By an approximation argument, it is sufficient to prove this conjecture for $n$-gons, where $n\geq 3$ is an arbitrary integer. We use polar coordinates, i.e., the vertices will be given as $(\varphi_{i},\varrho_{i})$, where $\varphi_{i}\in[0,2k\pi]$ — or $\varphi_{i}\in{\mathbb{R}}$ and $\varrho(\varphi)$ is $2k\pi$-periodic — and $\varrho_{i}\in(0,\infty)$. Since the central angles of the polygon are less than $\pi$, and their sum is $2k\pi$, therefore necessarily we have $n\geq 2k+1$.

Some numerical experimentation suggests for the case $k=2$ that in part 1) of the Conjecture we have actually a local minimum among pentagons, and for the case $k=3$ that in part 2) of the Conjecture we have actually a local minimum among octagons.

Theorem 9

Let $k\geq 2$ and $n\geq 2k+1$, and let $C\in{\Cal{C}}_{k}^{2}$ be a closed $n$-gon (degeneration to a polygon with less than $n$ vertices is excluded). Then we have $(1)\Longrightarrow(2)$, where

(1) $C$ is affinely equivalent to $C_{n,k}$,

(2) the volume product $V(C)V[\left(C-s(C)\right)^{*}]$ is a critical value.

For $n\geq 4k+1$ we have also $(2)\Longrightarrow(1)$.

Demonstration Proof

We begin with the proof of $(1)\Longrightarrow(2)$. For $C=C_{n,k}$ with $n\geq 2k+1$ an easy calculation shows that the partial derivatives of $V(C)V(C^{*})$ with respect to the angular and radial coordinates of the vertices are [math]. Observe that $C_{n,k}$ has an $n/(n,k)$-fold rotational symmetry (and $n/(n,k)\geq n/k\geq(2k+1)/k>2$), hence its Santaló point is [math]. Then applying the statement about the stability of the Santaló point, in [BMMR], Lemma 11 and [BM], Theorem E (valid also for ${\Cal{C}}^{2}_{k}$), we obtain statement (2) for $V(C)V[\left(C-s(C)\right)^{*}]$ rather than for $V(C)V(C^{*})$.

We turn to the proof of $(2)\Longrightarrow(1)$ for $n\geq 4k+1$. Observe that the average central angle of the sides is $2k\pi/n\leq 2k\pi/(4k+1)<\pi/2$. Hence the average sum of the central angles of two adjacent sides is at most $4k\pi/(4k+1)<\pi$. Therefore the sum of the central angles of some two adjacent sides is less than $\pi$. Using this, our proof for the usual case, i.e., for $k=1$, cf. [BM], Theorems A and F, gives that if $C$ gives a critical value of $V(C)V[\left(C-s(C)\right)^{*}]$, then some affine image of $C$ is inscribed to the unit circle about [math], has a positive orientation, and has equal sides. However, this polygonal line must close after $n$ steps, and just after a total angle of rotation $2k\pi$, hence it is $C_{n,k}$. $\blacksquare$

We can support our Conjecture 6 by investigating two special cases of it. Considering $C_{n,k}$ as inscribed to $S^{1}$, we can preserve the angular coordinates of the vertices of the conjectured $C_{n,k}$ while changing their radial coordinates, or we can preserve the radial coordinates of the vertices of the conjectured $C_{n,k}$ while changing their angular coordinates (and also their number). For the inverse Blaschke-Santaló inequality we give the exact lower bound of the volume product $V(C)V(C^{*})$ in the first case, for $2k+1\leq n\leq 4n$, and some positive bound in the second case, for $2k+1\leq n$.

Proposition 10

Let  $2k+1\leq n\leq 4k$. Let $C\in{\Cal{C}}_{k}^{2}$ be a closed $n$-gonal line with vertices having angular coordinates $2\pi ik/n\in[0,2k\pi]$, for $0\leq i\leq n$ (the [math]’th and $n$-th vertices coincide). Then $V(C)V(C^{*})\geq n^{2}\sin^{2}(k\pi/n)$, with equality only if either $C$ is a copy of $C_{n,k}$, magnified from the origin, or we have $n=4k$ and $C$ is a $k$ times traversed rhomb of centre [math].

Demonstration Proof

We write $\varrho(\cdot):[0,2k\pi]\to(0,\infty)$ for the radial function of $C$. Further, we write $\varphi_{i}:=2\pi ik/n$, and $\varrho_{i}:=\varrho(\varphi_{i})$. By $\varphi_{i}=i2k\pi/n$ the central angle of the side $[(\varphi_{i},\varrho_{i}),(\varphi_{i+1},\varrho_{i+1})]$ is $\vartheta_{i}=\vartheta:=2k\pi/n$. Then

[TABLE]

Now we are going to determine $V(C^{*})$. The angular domains, with vertices at [math] and with boundary rays passing through the vertices of $C$, decompose also $C^{*}$ into $n$ domains. Each of these domains $Q_{i}$ is a convex quadrangle, with one vertex at [math], and two sides beginning at [math], lying on the two boundary rays of the angular domain $(\varphi_{i},\varrho_{i})0(\varphi_{i+1},\varrho_{i+1})$. These two sides have lengths $1/\varrho_{i}$ and $1/\varrho_{i+1}$, and the other endpoints of these two sides have right angles in $Q_{i}$. The diagonal of $Q_{i}$ from [math] decomposes $Q_{i}$ into two right triangles, and $V(C^{*})$ is the sum of the areas of these two triangles. We may suppose that the angle bisector of the angle of $Q_{i}$ at [math] is the positive $x$-axis. Then we obtain by a elementary calculation that

[TABLE]

We write $F(\vartheta)$ and $G(\vartheta)$ for the coefficients of $1/\varrho_{i}^{2}+1/\varrho_{i+1}^{2}$ and $2\cdot(1/\varrho_{i})\cdot(1/\varrho_{i+1})$ in this formula. Then $F(\vartheta)\geq 0$: more exactly, for $n=4k$ we have $F(\vartheta)=0$, and for $2k+1\leq n\leq 4k-1$ we have $F(\vartheta)>0$. In fact, this last equality and inequality follow from

[TABLE]

and in the inequality here we have equality for $n=4k$, and strict inequality for $2k+1\leq n\leq 4k-1$.

By all these calculations we have

[TABLE]

Here the indices are considered cyclically, for both sums here, i.e., for $S_{1}:=V(C)$ and $S_{2}:=V(C^{*})$. Applying the arithmetic-geometric mean inequality both for $S_{1}$ and $S_{2}$, from A we obtain

[TABLE]

In B, in the arithmetic-geometric mean inequality for $S_{1}$ we have equality, if and only if all its summands are equal, i.e., if for each $i$ we have $\varrho_{i}\varrho_{i+1}=\varrho_{i+1}\varrho_{i+2}$, i.e.,

[TABLE]

(since $\varrho_{i+1}>0$). That is, for $n$ odd all $\varrho_{i}$’s are equal, in which case the statement of the theorem is proved, while for $n$ even the $\varrho_{i}$’s with $i$ of given parity are equal — i.e., the $\varrho_{i}$’s assume alternately two values. (The analogous consideration for $S_{2}$ will not be needed.)

In B, under the last product sign, again by the arithmetic-geometric mean inequality, we have for each $i$ that

[TABLE]

with equality for any $i$ if only if

[TABLE]

From B and D we obtain

[TABLE]

with equality only if both C and E hold. In other words, either $n=4k$, and $\varrho_{i}$’s assume alternately two values, or $2k+1\leq n\leq 4k-1$ and all $\varrho_{i}$’s are equal. In other words, we have $n=4k$ and $C$ is a $k$ times traversed rhomb of centre [math], thus is a non-singular linear image of $C_{n,k}=C_{4k,k}$ (in particular, $V(C)V(C^{*})=V(C_{4k,k})V(C_{4k,k}^{*})$), or $2k+1\leq n\leq 4k-1$ and all $\varrho_{i}$’s are equal, i.e., we have that $C$ is an inflation from [math] of $C_{n,k}$. $\blacksquare$

The following Proposition 12 proves Conjecture 6 in a special case, up to a constant factor about $0.43$. Before it we need a lemma.

Lemma 11

The functions $1/(1-\cos t)$ and $t/\sin t$ are strictly convex for $t\in(0,\pi)$.

Demonstration Proof

We begin with $1/(1-\cos t)$. Its derivative is

[TABLE]

which is strictly increasing since $\cos(t/2)$ is strictly decreasing and $\sin(t/2)$ is strictly increasing for $t\in(0,\pi)$.

Next we deal with $t/\sin t$. Its second derivative is

[TABLE]

and we have to show that here the numerator is positive. Equivalently, writing $s:=2t$,

[TABLE]

for $s=2t\in(0,2\pi)$. However, the left/right hand side of the last inequality is positive/non-positive for $s\in[\pi,2\pi)$, therefore the strict inequality holds here. Hence we suppose $s\in(0,\pi)$. Both sides of the last inequality are [math] for $s=0$, so it suffices to prove the respective inequality for the derivatives of the left and right hand side expressions. I.e., we have to prove

[TABLE]

Rearranging, this becomes

[TABLE]

which is valid for $s\in(0,\pi)$. $\blacksquare$

Proposition 12

Let $k\geq 2$ and $2k+1\leq n$ be integers. Let $C\in{\Cal{C}}_{k}^{2}$ be a closed polygonal line inscribed in the unit circle about [math]. Then $V(C)V(C^{*})\geq 4k^{2}/\min\{(4/\pi^{2})(t/\sin t)+2/(1-\cos t)\mid t\in(0,\pi)\}=k^{2}\cdot 1.7366\ldots$ .

Demonstration Proof

Let $C$ have $n$ vertices. Let the angles with vertex [math], spanned by the sides of the closed $n$-gonal line $C$, be $\vartheta_{i}\in(0,\pi)$, for $1\leq i\leq n$, where

[TABLE]

Then (for the first formula cf. the proof of Proposition 10),

[TABLE]

We choose some constant $c\in(0,\pi)$ (later we will optimize its value). Then

[TABLE]

We denote

[TABLE]

where (cf. A)

[TABLE]

For the first summand in C we use the estimates

[TABLE]

For the second summand in C we use the estimates

[TABLE]

Thus for the first summand in C we obtain by F

[TABLE]

Analogously, for the second summand in C we obtain by G, and by the arithmetic-harmonic mean inequality that

[TABLE]

Since each $\vartheta_{i}\in(c,\pi)$ is smaller than $\pi$, and their sum is $k_{2}\pi$, therefore for their number we obtain

[TABLE]

hence I gives

[TABLE]

By C, H and K we have

[TABLE]

(Observe that this holds also for $k_{1}=0$ and for $k_{2}=0$.)

Using E, we have $k_{1},k_{2}\in[0,2k]$, and we substitute $2k-k_{1}$ for $k_{2}$ in the last expression in L. Thus we obtain a quadratic polynomial $p(k_{1})$ of $k_{1}$. Next we minimize the value of $p(k_{1})$ for all $k_{1}\in[0,2k]$. Then clearly L will remain valid if we replace $p(k_{1})$ in the last expression of L by $\min\{p(k_{1})\mid k_{1}\in[0,2k]\}$. This minimum is attained for

[TABLE]

and its value is

[TABLE]

This value still depends on the arbitrarily chosen value $c\in(0,\pi)$. We have to choose $c$ so that N becomes maximum. Equivalently, we want to maximize the coefficient of $4k^{2}$ in N. It will be more convenient to consider the reciprocal of this coefficient, and then we will have to look for the minimum of this reciprocal. This reciprocal is

[TABLE]

By Lemma 11 O is a strictly convex function of $c\in(0,\pi)$. Moreover, it has limits at [math] and $\pi$ equal to $\infty$. Hence it has a unique minimum place $c_{0}$, which is the unique root of its derivative. That is,

[TABLE]

Solving this numerically, we find $c_{0}\approx 115.5^{\circ}$ and and the maximum value of N is $k^{2}\cdot 1.7366\ldots$, as asserted in the Proposition. $\blacksquare$

Definition Remark 12

Analogously to the planar case, possibly for each fixed $d$, for $C\in{\Cal{C}}^{d}_{k}$, there would hold a lower bound $V(C)V(C^{*})\geq{\text{const}}_{d}\cdot k^{2}$?

Definition Remark 13

For $d\geq 3$ and $k\geq 2$ possibly some polyhedral surfaces would minimize $V(C)V(C^{*})$ in ${\Cal{C}}^{d}_{k}$. For $\Sigma$ a simplex of barycentre [math], suitable maps ${\text{bd}}\,\Sigma\to{\text{bd}}\,\Sigma$ of index $k$ (analogous to maps $S^{d-1}\to S^{d-1}$ of index $k$) give $V(C)V(C^{*})=k^{2}(d+1)^{d+1}/(d!)^{2}$. In particular, for $d=3$ they give $(64/9)k^{2}$. However, a right prism of height $2$ over the planar conjectured extremal curve, realized as a direct product with $[-1,1]$, with polar the respective bipyramid of height $2$, gives for $k\geq 3$ better, while for $k=2$ the doubly traversed simplex gives better. In fact, by Remark 7, this inequality is

[TABLE]

This can be rewritten as

[TABLE]

Here the right hand side is greater than $1$ for all $k\geq 4$, so in this case this inequality holds. For $k=3$ a direct calculation shows the same. However, for $k=2$ the converse inequality holds. For higher dimensions, one could take products of lower dimensional examples, with the indices being multiplied (with the $l^{\infty}$- or the $l^{1}$-norm), iteratedly. However, we do not have a reasonable conjecture for $C\in{\Cal{C}}^{d}_{k}$, for any given $d\geq 3$ and $k\geq 2$, except possibly for $d=3$ and $k=2$ the doubly traversed simplex?