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Pál's isominwidth inequality for ball convex bodies in planes of constant curvature

Ferenc Fodor, Nathan Robock, Ádám Sagmeister

Abstract

Pál's classical isominwidth inequality states that the regular triangle has minimal area among plane convex bodies of minimal width $w$. A similar result is the Blaschke--Lebesgue inequality that states that Reuleaux triangles minimize the area among bodies of constant width $w$ in the plane. In this paper, we connect these two problems by solving the isominwidth problem for $r$-ball convex bodies in the Euclidean, hyperbolic and spherical planes.

Pál's isominwidth inequality for ball convex bodies in planes of constant curvature

Abstract

Pál's classical isominwidth inequality states that the regular triangle has minimal area among plane convex bodies of minimal width . A similar result is the Blaschke--Lebesgue inequality that states that Reuleaux triangles minimize the area among bodies of constant width in the plane. In this paper, we connect these two problems by solving the isominwidth problem for -ball convex bodies in the Euclidean, hyperbolic and spherical planes.
Paper Structure (2 sections, 2 theorems, 2 equations)

This paper contains 2 sections, 2 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries

Key Result

Theorem 1.1

Let $K$ be an $r$-ball convex body in $\mathcal{M}^2$ where $\mathcal{M}^2$ stands for one of ${\mathbb R}^2$, ${\mathbb H}^2$ and ${\mathbb S}^2$. Let $w$ denote the minimal width of $K$, and let $w\leq r$ (in the case $\mathcal{M}^2={\mathbb S}^2$ we also assume $r<\frac{\pi}{2}$). Then with equality if and only if $K$ is congruent to $T_{w,r}$.

Theorems & Definitions (2)

  • Theorem 1.1
  • Theorem 1.2