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 $\boldsymbol{n}_1,\dots,\boldsymbol{n}_5$ 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.