A Positive Answer to B\'ar\'any's Question on Face Numbers of Polytopes

TL;DR

This paper affirms Bárány's question on face numbers of convex polytopes, providing explicit lower bounds for face counts relative to vertices and facets, with characterizations of equality cases.

Contribution

It proves a stronger inequality for face numbers of convex polytopes, extending previous understanding and characterizing cases of equality.

Findings

Established lower bounds for face numbers relative to vertices and facets.

Characterized when equality holds in the bounds for simple and simplicial polytopes.

Confirmed Bárány's conjecture affirmatively for all convex polytopes.

Abstract

Despite a full characterization of the face vectors of simple and simplicial polytopes, the face numbers of general polytopes are poorly understood. Around 1997, B\'ar\'any asked whether for all convex $d$-polytopes $P$ and all $0 \leq k \leq d-1$, $f_k(P) \geq \min\{f_0(P), f_{d-1}(P)\}$. We answer B\'ar\'any's question in the affirmative and prove a stronger statement: for all convex $d$-polytopes $P$ and all $0 \leq k \leq d-1$, \[ \frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} + {\lfloor \frac{d}{2} \rfloor \choose k}\biggr], \qquad \frac{f_k(P)}{f_{d-1}(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose d-k-1} + {\lfloor \frac{d}{2} \rfloor \choose d-k-1}\biggr]. \] In the former, equality holds precisely when $k=0$ or when $k=1$ and $P$ is simple. In the latter, equality holds precisely when $k=d-1$ or when $k=d-2$ and $P$ is simplicial.