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

Abstract

We choose some special unit vectors ${\mathbf{n}}_1,\dots ,{\mathbf{n}}_5$ in ${\mathbb{R}}^3$ and denote by $\mathcal{L}\subset {\mathbb{R}}^5$ 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 ${\mathbf{n}}_1,\dots ,{\mathbf{n}}_5$ (and no other vector) are unit outward normals to the faces of $P$ and the perimeter of the face with the outward normal ${\mathbf{n}}_k$ is equal to $L_k$ for all $k=1,\dots ,5$. Our main result reads that $\mathcal{L}$ is not a locally-analytic set, i.\,e., we prove that, for some point $(L_1,\dots ,L_5)\in \mathcal{L}$, it is not possible to find a neighborhood $U\subset {\mathbb{R}}^5$ and an analytic set $A\subset {\mathbb{R}}^5$ such that $\mathcal{L}\cap U=A\cap U$. 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.

arXiv comment9 pages, 3 figures. In version 2, some misprints are corrected