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

Article

## 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.

## Keywords

Euclidean space Convex polyhedron Perimeter of a face Analytic set

52B10 51M20

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