On variational problems related to steepest descent curves and self dual convex sets on the sphere

TL;DR

This paper investigates variational problems involving convex sets on the sphere, identifying minimal configurations with constant width and specific geometric properties, and relates these to steepest descent curves for quasi-convex functions.

Contribution

The paper introduces a new variational problem on the sphere involving dual convex sets and characterizes the minimizers, including the Reuleaux triangle in three dimensions, extending previous planar results.

Findings

Minima are sets of constant width π/2 with boundary points on the dual set.

In three dimensions, the minimal set is a Reuleaux triangle on the sphere.

Results differ significantly for dimensions three and higher.

Abstract

Let $\mathcal{C}$ be the family of compact convex subsets $S$ of the hemisphere in $\rn$ with the property that $S$ contains its dual $S^*;$ let $u\in S^*$, and let $ \Phi(S,u)=\frac{2}{\omega_n}\int_{S}\ < \theta, u \ > \,\, d\sigma(\theta). $ The problem to study $ \inf \big\{\Phi(S,u), S \in \mathcal{C}, \, u\in S^* \big\} $ is considered. It is proved that the minima of $ \Phi $ are sets of constant width $ \pi/2 $ with $ u $ on their boundary. More can be said for $n=3$: the minimum set is a Reuleaux triangle on the sphere. The previous problem is related to the one to find the maximal length of steepest descent curves for quasi convex functions, satisfying suitable constraints. For $ n=2 $ let us refer to \cite{Manselli-Pucci}. Here quite different results are obtained for $ n\geq 3$.