Steepest descent curves of convex functions on surfaces of constant curvature

TL;DR

This paper establishes upper bounds on the length of steepest descent curves of quasi-convex functions on surfaces of constant curvature, generalizing Euclidean results and demonstrating optimality on the sphere.

Contribution

It extends the theory of steepest descent curves to spherical and hyperbolic surfaces, introducing G-curves and proving their properties and optimal bounds.

Findings

Length of steepest descent curves is bounded by the perimeter of the convex set.

Existence of G-curves with length equal to the convex hull perimeter on the sphere.

Generalization of Euclidean theorems to curved surfaces.

Abstract

Let S be a complete surface of constant curvature K = + 1 or -1, i.e. the sphere S^2 or the Lobachevskij plane L^2, and D a bounded convex subset of S. If S = S^2, assume also diameter (D) < pi/2. It is proved that the length of any steepest descent curve of a quasi-convex function in D is less than or equal to the perimeter of D. This upper bound is actually proved for the class of G-curves, a family of curves that naturally includes all steepest descent curves. In case S = S^2, it is also proved the existence of G-curves, whose length is equal to the perimeter of their convex hull, showing that the above estimate is indeed optimal. The results generalize theorems by Manselli and Pucci on steepest descent curves in the Euclidean plane.