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 ${\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5$ in ${\mathbb {R}}^3$ and denote by ${\mathscr {L}}\subset {\mathbb {R}}^5$ the set of all points $(L_1,\ldots ,L_5)\in {\mathbb {R}}^5$ with the following property: there exists a compact convex polytope $P\subset {\mathbb {R}}^3$ such that the vectors ${\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5$ (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 $k=1,\ldots ,5$. Our main result reads that ${\mathscr {L}}$ is not a locally-analytic set, i.e., we prove that, for some point $(L_1,\ldots ,L_5)\in {\mathscr {L}}$, 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.