Compact convex sets with prescribed facial dimensions

TL;DR

This paper investigates the possible face dimension patterns of general compact convex sets, showing that any finite sequence of face dimensions can be realized, and explores complex face structures including fractal dimensions.

Contribution

It demonstrates that for any finite sequence of positive integers, there exist compact convex sets with faces only of those dimensions, extending understanding beyond polytopes.

Findings

Existence of convex sets with prescribed face dimension sequences

Examples of convex sets with fractal dimension of face unions

Discussion of nontrivial face dimensionality patterns

Abstract

While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the faces of general closed convex sets. We show that for any finite sequence of positive integers there exist compact convex sets which only have extreme points and faces with dimensions from this prescribed sequence. We also discuss another approach to dimensionality, considering the dimension of the union of all faces of the same dimension. We show that the questions arising from this approach are highly nontrivial and give examples of convex sets for which the sets of extreme points have fractal dimension.