2023/12/07 by Vera Roshchina, Roshchina, Vera, Levent Tunçel +1
Biochemistry, Genetics and Molecular Biology · Materials Science · Neuroscience · #52A38 #90C22 #90C25 #Combinatorics (math.CO) #FOS: Mathematics #Nuclear Receptors and Signaling #Optimization and Control (math.OC) #Photochromic and Fluorescence Chemistry #Retinal Development and Disorders
paper · pdf · doi:10.48550/arxiv.2312.04419
openalex publication_date 2023/12/07 · openalex created_date 2023/12/09 · openalex updated_date 2026/07/28
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.