Stability of $C^\infty$ convex integrands

TL;DR

This paper proves that stable smooth convex integrands form an open and dense subset within all smooth convex integrands, highlighting their generic stability in the Whitney topology.

Contribution

It establishes the openness and density of stable convex integrands in the space of all smooth convex integrands, providing a foundational result in convex analysis.

Findings

Stable convex integrands are open in the Whitney topology.

Stable convex integrands are dense among smooth convex integrands.

Application of the main result demonstrates practical implications.

Abstract

In this paper, it is shown that the set consisting of stable convex integrands $S^n\to \mathbb{R}_+$ is open and dense in the set consisting of $C^\infty$ convex integrands with respect to Whitney $C^\infty$ topology. Moreover, an application of the proof of this result is also shown.