The isominwidth problem on the 2-sphere

TL;DR

This paper investigates the spherical isominwidth problem, identifying the minimizers of area for different width ranges on the 2-sphere and establishing stability versions of the inequalities.

Contribution

It extends the spherical isominwidth theorem by characterizing minimizers for widths greater than π/2 and provides stability results for these inequalities.

Findings

Regular spherical triangles minimize area for width ≤ π/2.

Polar sets of spherical Reuleaux triangles minimize area for width > π/2.

Stability versions of the inequalities are established.

Abstract

P\'al's isominwidth theorem states that for a fixed minimal width, the regular triangle has minimal area. A spherical version of this theorem was proven by Bezdek and Blekherman, if the minimal width is at most $\tfrac \pi 2$. If the width is greater than $\tfrac \pi 2$, the regular triangle no longer minimizes the area at fixed minimal width. We show that the minimizers are instead given by the polar sets of spherical Reuleaux triangles. Moreover, stability versions of the two spherical inequalities are obtained.