Existence of polytopal local maximizers with the structural properties used in the main theorems

Ascertain whether there exist polytopes in R^n that locally maximize the isotropic constant and possess the specific boundary structures considered in the paper, including: (i) having a simplicial vertex; (ii) being centrally symmetric and having a simplicial vertex while maximizing within the class of centrally symmetric convex bodies; and (iii) being a zonotope that locally maximizes within the centrally symmetric class and has a cubical zone.

Background

The paper proves several conditional results showing that if a local maximizer of the isotropic constant has certain boundary structures, then it must be one of the high-dimensional “Platonic solids” (simplex, cross-polytope, or cube).

However, the authors note that these results are hypothetical in the sense that it is currently unknown whether local maximizers with these specific structural properties exist at all. Establishing existence (or non-existence) of such polytopal local maximizers would clarify the applicability of these structural theorems.