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

Abstract

Let be the family of compact convex subsets of the hemisphere in with the property that contains its dual let , and let The problem to study is considered. It is proved that there exists a minimizing couple such that is self-dual and is on its boundary. More can be said for : 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 , let us refer to [Manselli P, Pucci C. Maximum length of steepest descent curves for quasi-convex functions. Geom. Dedicata. 1991]. Here, quite different results are obtained for .

Keywords:

• convex sets
• constant width on the sphere
• steepest descent curves
• self-dual cones

AMS Subject Classifications:

• Primary: 52A20
• Secondary: 52A10
• 52A38
• 49K30