We choose some special unit vectors $${\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5$$n1,…,n5 in $${\mathbb {R}}^3$$R3 and denote by $${\mathscr {L}}\subset {\mathbb {R}}^5$$L⊂R5 the set of all points $$(L_1,\ldots ,L_5)\in {\mathbb {R}}^5$$(L1,…,L5)∈R5 with the following property: there exists a compact convex polytope $$P\subset {\mathbb {R}}^3$$P⊂R3 such that the vectors $${\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5$$n1,…,n5 (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal $${\mathbf {n}}_k$$nk is equal to $$L_k$$Lk for all $$k=1,\ldots ,5$$k=1,…,5. Our main result reads that $${\mathscr {L}}$$L is not a locally-analytic set, i.e., we prove that, for some point $$(L_1,\ldots ,L_5)\in {\mathscr {L}}$$(L1,…,L5)∈L, it is not possible to find a neighborhood $$U\subset {\mathbb {R}}^5$$U⊂R5 and an analytic set $$A\subset {\mathbb {R}}^5$$A⊂R5 such that $${\mathscr {L}}\cap U=A\cap U$$L∩U=A∩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.

中文翻译：

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为什么具有指定方向和面周长的凸多面体没有存在定理？

我们选择一些特殊的单位向量 $${\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5$$n1,...,n5 在 $${\mathbb {R}}^3$$R3并用 $${\mathscr {L}}\subset {\mathbb {R}}^5$$L⊂R5 表示所有点的集合 $$(L_1,\ldots ,L_5)\in {\mathbb {R }}^5$$(L1,…,L5)∈R5 具有以下性质：存在紧凸多面体 $$P\subset {\mathbb {R}}^3$$P⊂R3 使得向量 $ ${\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5$$n1,...,n5（没有其他向量）是 P 的面的单位外法线和面的周长外向法线 $${\mathbf {n}}_k$$nk 等于 $$L_k$$Lk 所有 $$k=1,\ldots ,5$$k=1,...,5。我们的主要结果表明 $${\mathscr {L}}$$L 不是局部解析集，即我们证明，对于某个点 $$(L_1,\ldots ,L_5)\in {\mathscr { L}}$$(L1,…,L5)∈L，不可能找到邻域 $$U\subset {\mathbb {R}}^5$$U⊂R5 和解析集 $$A\subset {\mathbb {R}}^5$$A⊂R5使得 $${\mathscr {L}}\cap U=A\cap U$$L∩U=A∩U。我们将此结果解释为寻找具有指定方向和面周长的紧凑凸多面体的存在定理的障碍。