一切皆有可能：构造具有指定面维数的谱面体
Everything is possible: constructing spectrahedra with prescribed facial dimensions
AI总结：
本文证明任意有限非负整数集均可作为闭凸集的面维数特征，且该集合可实现为凸二次不等式系统解集（即谱面体），同时给出了所需最少不等式数的下界并探讨其与整数序列不可分解性的联系。
摘要：
给定任意有限非负整数集，存在一个闭凸集，其面维数特征与该整数集一致，即其非空面的维数恰好构成该整数集。在本文中，我们证明此类集合可以实现为有限多个凸二次不等式系统的解集，因此可通过二阶锥规划问题表示，且特别地，它是谱面体。由此还可得出，与早期构造不同，这些集合是面暴露的。我们得到了表示具有指定面维数特征的闭凸集所需的最少凸二次不等式数量的下界，并证明该界在某些特殊情况下是紧的。最后，我们将寻找有效表示的问题与整数序列的不可分解性及其他主题联系起来，并讨论了大量未决问题。
英文摘要：
Given any finite set of nonnegative integers, there exists a closed convex set whose facial dimension signature coincides with this set of integers, that is, the dimensions of its nonempty faces comprise exactly this set of integers. In this work, we show that such sets can be realised as solution sets of systems of finitely many convex quadratic inequalities, and hence are representable via second-order cone programming problems, and are, in particular, spectrahedral. It also follows that these sets are facially exposed, in contrast to earlier constructions. We obtain a lower bound on the minimum number of convex quadratic inequalities needed to represent a closed convex set with prescribed facial dimension signature, and show that our bound is tight for some special cases. Finally, we relate the question of finding efficient representations with indecomposability of integer sequences and other topics, and discuss a substantial number of open questions.
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。
