一个丢失的反例和关于有光照多面体的一个问题
A Lost Counterexample and a Problem on Illuminated Polytopes
摘要：
在1984年的一篇论文中，丹尼尔·A.马库斯声称有一个反例来证明他的猜想，即R^m中的最小正k-生成向量配置的大小至多为2 km。然而，反例从未发表，似乎已经丢失。1974年，彼得·马尼（Peter Mani）独立解决了哈德维格（Hadwiger）的一个问题，证明了每个“发光”的d维多面体必须至少有2d顶点。这两个研究是由盖尔对偶，一个基本的线性代数技术设计的Micha A。60年代的Perles因此，我们注意到，马尼的研究提供了一个反例马库斯猜想的参数，正是马库斯声称。在另一个方向上，与马库斯的工具，我们提供了一个答案，一个问题留下开放的马尼：可以“照明”d维多面体上的最小数量的顶点是非单纯的？
英文摘要：
In a Note added in proof to a 1984 paper, Daniel A. Marcus claimed to have a counterexample to his conjecture that a minimal positively k-spanning vector configuration in R^m has size at most 2km. However, the counterexample was never published, and seems to be lost. Independently, ten years earlier, Peter Mani in 1974 solved a problem by Hadwiger, disproving that every ``illuminated'' d-dimensional polytope must have at least 2d vertices. These two studies are related by Gale duality, an elementary linear algebra technique devised by Micha A. Perles in the sixties. Thus, we note that Mani's study provides a counterexample for Marcus' conjecture with exactly the parameters that Marcus had claimed. In the other direction, with Marcus' tools we provide an answer to a problem left open by Mani: Could ``illuminated'' d-dimensional polytopes on a minimal number of vertices be nonsimplicial?
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。
