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

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 $S^{\ast };$ let $u\in S^{\ast }$, and let $\Phi (S,u)=\frac{2}{{\omega }_n}{\int }_S<\theta ,u>d\sigma (\theta ).$ The problem to study $\inf \{\Phi (S,u),S\in \mathcal{C},u\in S^{\ast }\}$ 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 $n\ge 3$.

13 pages, 1 figure