为什么没有一个凸多面体的存在性定理,它的面有规定的方向和周长?

IF 0.4 4区 数学 Q4 MATHEMATICS
Victor Alexandrov
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引用次数: 1

摘要

我们在({\mathbb{R}}^3\)中选择一些特殊的单位向量\({\ mathbf{n}{_1,\ldots,{\math bf{n}}_5\),并用({\smathscr{L})\subet{\mattbb{R}}}^5\)表示所有点\(((L_1,\ldot,L_5)\在{\mastbb{R}^5 \)中的集合,其性质如下:存在一个紧致凸多面体\(P\subet}_1,\ldots,{\mathbf{n}}_5)(并且没有其他向量)是P的面的单位向外法线,并且具有向外法线的面的周长\({\mathbf{n}}_k\)对于所有\(k=1,\ldots,5\)等于\(L_k\)。我们的主要结果表明,\({\mathscr{L}})不是局部分析集,即,我们证明了,对于{\math scr{L}}中的某个点\((L_1,\ldots,L_5)\),不可能找到邻域\(U\subet{\matthbb{R})^5\和分析集\(a\subet{\mathbb{R}}^5\),使得\。我们将这一结果解释为寻找具有指定方向和面周长的紧致凸多面体的存在性定理的障碍。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?

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.

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来源期刊
CiteScore
0.80
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.
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