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Let $\mathcal{C}$ be the family of compact convex subsets $S$ of thehemisphere 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 theminima of $ \Phi $ are sets of constant width $ \pi/2 $ with $ u $ on theirboundary. More can be said for $n=3$: the minimum set is a Reuleaux triangle onthe sphere. The previous problem is related to the one to find the maximallength of steepest descent curves for quasi convex functions, satisfyingsuitable constraints. For $ n=2 $ let us refer to \cite{Manselli-Pucci}. Herequite different results are obtained for $ n\geq 3$.

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