P\'al's isominwidth inequality for ball convex bodies in planes of constant curvature

TL;DR

This paper extends classical isominwidth inequalities to r-ball convex bodies across Euclidean, hyperbolic, and spherical planes, unifying and generalizing known results about minimal area shapes.

Contribution

It solves the isominwidth problem for r-ball convex bodies in various constant curvature planes, linking classical inequalities in a unified framework.

Findings

Regular triangle minimizes area among convex bodies of minimal width in Euclidean plane.

Reuleaux triangles minimize area among bodies of constant width in Euclidean plane.

The results are extended to hyperbolic and spherical geometries.

Abstract

P\'al'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.