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Asymptotics for face numbers of certain Hanner polytopes, with applications

Tomer Milo

Abstract

We provide asymptotics for the number of faces of a certain family of Hanner polytopes. As a corollary, we come close to saturating the FLM inequality for a certain family of parameters.

Asymptotics for face numbers of certain Hanner polytopes, with applications

Abstract

We provide asymptotics for the number of faces of a certain family of Hanner polytopes. As a corollary, we come close to saturating the FLM inequality for a certain family of parameters.
Paper Structure (6 sections, 18 theorems, 66 equations)

This paper contains 6 sections, 18 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Face numbers as a recursion sequence
  3. Solving the recursion for $H_m$ using trees
  4. The Upper Bound
  5. The Lower Bound
  6. Extending to all dimensions, and proof of the main Theorems

Key Result

Theorem 1.1

Let $P \subset \mathbb R^n$ be a polytope. Let $V$, $\mathcal{F}$ denote the set vertices $(0-\text{dimensional faces})$ and facets $((n-1)-\text{dimensional faces})$ respectively. Let $B_2^n$ denote the Euclidean unit ball, and assume $rB_2^n \subset P \subset RB_2^n$. Then where $c>0$ is some universal constant.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proposition 3.1
  • ...and 22 more