The sparseness of g-convex functions

TL;DR

This paper investigates the properties of g-convex functions on manifolds, establishing criteria for their existence and demonstrating their sparseness in various function spaces, which impacts differential geometry and optimization.

Contribution

It provides new criteria for the existence of metrics making functions g-convex and shows that g-convex functions are generally sparse in smooth function spaces.

Findings

The set of g-convex functions on a compact manifold is nowhere dense.

Most g-convex polynomials on Rn have at most one critical point.

Density of g-convex univariate and specific polynomial classes diminishes to zero.

Abstract

The g-convexity of functions on manifolds is a generalization of the convexity of functions on Rn. It plays an essential role in both differential geometry and non-convex optimization theory. This paper is concerned with g-convex smooth functions on manifolds. We establish criteria for the existence of a Riemannian metric (or connection) with respect to which a given function is g-convex. Using these criteria, we obtain three sparseness results for g-convex functions: (1) The set of g-convex functions on a compact manifold is nowhere dense in the space of smooth functions. (2) Most polynomials on Rn that is g-convex with respect to some geodesically complete connection has at most one critical point. (3) The density of g-convex univariate (resp. quadratic, monomial, additively separable) polynomials asymptotically decreases to zero