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A very short proof of the Figiel-Lindenstrauss-Milman theorem

Tomer Milo

Abstract

We provide a short proof for the Figiel, Lindenstrauss and Milman inequality regarding the number of vertices and faces of certain polytope, with an explicit bound on the universal constant involved. The proof is completely elementary and avoids any form of Dvoretzky's theorem, as well as the spherical isoperimetric inequality.

A very short proof of the Figiel-Lindenstrauss-Milman theorem

Abstract

We provide a short proof for the Figiel, Lindenstrauss and Milman inequality regarding the number of vertices and faces of certain polytope, with an explicit bound on the universal constant involved. The proof is completely elementary and avoids any form of Dvoretzky's theorem, as well as the spherical isoperimetric inequality.

Paper Structure

This paper contains 4 theorems, 8 equations.

Key Result

Theorem 1

There exists an absolute constant $c>0$ such that for every $n \in \mathbb{N}$ and for every convex polytope $P \subset \mathbb{R}^{n}$ which is symmetric about the origin (that is, $P= -P$) we have: Where $V$ and $\mathcal{F}$ are the sets of vertices ($0$-dimensional faces) and facets ($(n-1)$-dimensional faces) of $P$, respectively. Moreover, the same estimate holds for a general polytope $P$

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • proof
  • proof : Proof of Theorem \ref{['main theorem']}