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Rogers's proof of Vaaler's theorem

Roman Karasev

Abstract

We note that an argument by Rogers (1958) gives a proof of Vaaler's theorem (1979) about sections of the cube and allows certain generalizations of the theorem.

Rogers's proof of Vaaler's theorem

Abstract

We note that an argument by Rogers (1958) gives a proof of Vaaler's theorem (1979) about sections of the cube and allows certain generalizations of the theorem.

Paper Structure

This paper contains 3 sections, 5 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Proof of the volume estimate
  3. Proof of the surface area estimate

Key Result

Theorem 1.1

Let $P\subset\mathbb R^n$ be a polyhedron containing the origin in its interior and having the property that for any face $F\subset P$ of codimension $k$ the distance from $0$ to the affine span of $F$ is at least $\sqrt k$. Then the volume of $P$ is at least $2^n$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • proof : Deduction of Vaaler's theorem from Theorem \ref{['theorem:rogers']}
  • proof : Proof of Theorem \ref{['theorem:rogers']}, following the proof in rogers1958
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4: Given only to complete the above remark
  • proof
  • ...and 3 more