Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?

TL;DR

This paper demonstrates that the set of possible face perimeters for convex polytopes with fixed face directions in three dimensions is not locally analytic, indicating fundamental obstacles to formulating an existence theorem for such polytopes.

Contribution

It proves that the set of feasible face perimeters is not locally analytic, revealing intrinsic limitations in characterizing convex polytopes with prescribed face directions and perimeters.

Findings

The set of feasible face perimeters is not a locally-analytic set.

This non-analyticity presents obstacles to existence theorems.

The result applies to specific fixed face directions in b2^3.

Abstract

We choose some special unit vectors $\boldsymbol{n}_1,\dots,\boldsymbol{n}_5$ in $\mathbb{R}^3$ and denote by $\mathscr{L}\subset\mathbb{R}^5$ the set of all points $(L_1,\dots,L_5)\in\mathbb{R}^5$ with the following property: there exists a compact convex polytope $P\subset\mathbb{R}^3$ such that the vectors $\boldsymbol{n}_1,\dots,\boldsymbol{n}_5$ (and no other vector) are unit outward normals to the faces of $P$ and the perimeter of the face with the outward normal $\boldsymbol{n}_k$ is equal to $L_k$ for all $k=1,\dots,5$. Our main result reads that $\mathscr{L}$ is not a locally-analytic set, i.\,e., we prove that, for some point $(L_1,\dots,L_5)\in\mathscr{L}$, it is not possible to find a neighborhood $U\subset\mathbb{R}^5$ and an analytic set $A\subset\mathbb{R}^5$ such that $\mathscr{L}\cap U=A\cap U$. We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces.