Complexity yardsticks for $f$-vectors of polytopes and spheres

TL;DR

This paper investigates measures of complexity for $f$-vectors of polytopes and spheres, highlighting differences between simplicial and general polytopes and between various types of CW complexes in higher dimensions.

Contribution

It introduces geometric and computational complexity measures to distinguish $f$-vector types of polytopes and spheres, emphasizing qualitative differences in higher dimensions.

Findings

Identifies qualitative differences between $f$-vectors of simplicial and general polytopes.

Highlights distinctions between flag $f$-vectors of polytopes and CW spheres.

Proposes complexity measures to differentiate these structures.

Abstract

We consider geometric and computational measures of complexity for sets of integer vectors, asking for a qualitative difference between $f$-vectors of simplicial and general $d$-polytopes, as well as flag $f$-vectors of $d$-polytopes and regular CW $(d-1)$-spheres, for $d\ge 4$.