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Small volume bodies of constant width

Andrii Arman, Andriy Bondarenko, Fedor Nazarov, Andriy Prymak, Danylo Radchenko

TL;DR

The paper addresses the existence of small-volume convex bodies of constant width in high dimensions. It constructs an explicit constant-width-2 body $M$ as an intersection of translates of the ball, proves $M$ has constant width, and derives a nontrivial upper bound on ${\rm Vol}(M)$ that decays like $0.9^n$ times ${\rm Vol}(B^n)$ for large $n$. The core approach partitions volume by orthants and uses a containing triangle to bound the mixed-orthant contributions, obtaining a numeric bound $s<1.8$ that yields $r_n < 0.9$ for large $n$. This answers Schramm's question affirmatively and provides a constructive, dimension-dependent decrease in volume with a finite verification for small $n$.

Abstract

For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of O.~Schramm.

Small volume bodies of constant width

TL;DR

The paper addresses the existence of small-volume convex bodies of constant width in high dimensions. It constructs an explicit constant-width-2 body as an intersection of translates of the ball, proves has constant width, and derives a nontrivial upper bound on that decays like times for large . The core approach partitions volume by orthants and uses a containing triangle to bound the mixed-orthant contributions, obtaining a numeric bound that yields for large . This answers Schramm's question affirmatively and provides a constructive, dimension-dependent decrease in volume with a finite verification for small .

Abstract

For every large enough , we explicitly construct a body of constant width that has volume less than ), where is the unit ball in . This answers a question of O.~Schramm.
Paper Structure (3 sections, 2 theorems, 31 equations, 3 figures)

This paper contains 3 sections, 2 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and the construction of $M$
  3. Estimate for ${\rm Vol}(M)$

Key Result

Theorem 1

For every sufficiently large $n$, there is a body $M$ in ${\mathbb{R}}^n$ of constant width $2$ with ${\rm Vol}(M)< 0.9^n {\rm Vol}({\mathbb{B}}^{n})$.

Figures (3)

  • Figure 1: Illustration of the body $M$ in dimensions $n=2,3$.
  • Figure 2: Sets $A$, $A'$, and a Reuleaux triange $\mathcal{R}$ that contains both $A$ and $A'$.
  • Figure 3: Triangle $T_{\alpha,\beta}$ that contains $A$.

Theorems & Definitions (12)

  • Theorem 1
  • Claim 2
  • proof
  • Claim 3
  • proof
  • Remark 1
  • Corollary 4
  • proof
  • Claim 5
  • proof
  • ...and 2 more