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

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 $n\geq 3$.