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

Abstract

Let C be the family of compact convex subsets S of the hemisphere in R n with the property that S contains its dual S * ; let u ∈ S * , and let (S, u) = 2 ω n S θ, u dσ (θ). The problem to study inf (S, u), S ∈ C, u ∈ S * is considered. It is proved that there exists a minimizing couple (S, u) such that S is self-dual and u is on its 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 [Manselli P, Pucci C. Maximum length of steepest descent curves for quasi-convex functions. Geom. Dedicata. 1991]. Here, quite different results are obtained for n ≥ 3.

On steepest descent curves for quasi convex families in Rn

“…in . The minimum of the integral on the left in has been investigated in . Remark The property , together with the property (i) of Lemma are necessary for a curve γ to be a SEC. In the Example it is noticed that these properties are not sufficient.…”

“…Existence, uniqueness and stability results are proved for in Theorems , . The results of this part are aimed to avoid assumptions of regularity for f in and to obtain a compact class (the expanding couples) for studying suitable variational problems as in . Also it is proved that, if is an EC and , then the length of the part of γ between Ω 1 and Ω 2 satisfies the bound: with Hausdorff distance and constant depending only on the dimension n (see Theorem ).…”

Bounding Regions to Plane Steepest Descent Curves of Quasiconvex Families

“…For n = 2, ϑ ∈ R, let θ = (cos ϑ, sin ϑ) ∈ S 1 and h K (ϑ) := H K (θ), it will be denoted h(ϑ) if no ambiguity arises. For every θ ∈ S 1 there exists at least one point x ∈ ∂K such that: θ, y − x ≤ 0 ∀y ∈ K; (10) this means that the line through x orthogonal to θ supports K. For every x ∈ ∂K let N x the set of θ ∈ S 1 such that (10) holds. Let F (θ) be the set of all x ∈ ∂K satisfying (10).…”

“…For every θ ∈ S 1 there exists at least one point x ∈ ∂K such that: θ, y − x ≤ 0 ∀y ∈ K; (10) this means that the line through x orthogonal to θ supports K. For every x ∈ ∂K let N x the set of θ ∈ S 1 such that (10) holds. Let F (θ) be the set of all x ∈ ∂K satisfying (10). If ∂K is strictly convex at the direction θ then F (θ) reduces to one point and it will be denoted by x(θ).…”

“…Sharp bounds about the length of the steepest descent curves for a quasi convex family, have been proved in [8], [11], [12]. The geometry of these curves, equivalent definitions, related questions and generalizations have been studied in [1], [3], [4], [9], [10].…”