包含与内接单形
Containment and inscribed simplices
摘要：
设K和L是R^n中的紧凸集.证明了以下两个命题是等价的：（i）对于K中的每个至多有n+1个顶点的多面体Q，L包含Q的平移。(ii)L包含K的翻译。设1 <= d <= n-1。证明了以下两个命题是等价的：（i）对于K内至多d+1个顶点的多面体Q，L包含Q的平移. (ii)对于每个d维子空间W，集合L在W上的正交投影包含集合K在W上的相应投影的平移。证明了：若K是R^n中至少有d+2个暴露点的紧凸集，则存在紧凸集L，使得L的每个d维正交投影都包含K的相应投影的平移，而L不包含K的平移.特别地，只要dim（K）> d，就存在这样的凸体L。
英文摘要：
Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 <= d <= n-1. It is also shown that the following two statements are equivalent: (i) For every polytope Q inside K having at most d+1 vertices, L contains a translate of Q. (ii) For every d-dimensional subspace W, the orthogonal projection of the set L onto W contains a translate of the corresponding projection of the set K onto W. It is then shown that, if K is a compact convex set in R^n having at least d+2 exposed points, then there exists a compact convex set L such that every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K, while L does not contain a translate of K. In particular, such a convex body L exists whenever dim(K) > d.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
