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Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces? (English) Zbl 1392.52007
Summary: 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.

##### MSC:
 52B10 Three-dimensional polytopes 51M20 Polyhedra and polytopes; regular figures, division of spaces
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##### References:
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