Simultaneous stability of $C^\infty$ convex integrands and their duals

TL;DR

This paper proves that the dual of a smooth or stable convex integrand on the sphere retains the same properties, and establishes a Morse index relationship between critical points of a convex integrand and its dual.

Contribution

It demonstrates the simultaneous stability and smoothness of convex integrands and their duals, and relates their critical points via Morse indices.

Findings

Dual of a $C^ abla$ convex integrand is $C^ abla$.

Stability of a convex integrand implies stability of its dual.

Critical points of a convex integrand correspond to dual critical points with complementary Morse indices.

Abstract

In this paper, the following three are shown. (1) For a $C^\infty$ convex integrand $\gamma: S^n\to \mathbb{R}_+$, its dual convex integrand $\delta: S^n\to \mathbb{R}_+$ is of class $C^\infty$. (2) For a stable convex integrand $\gamma: S^n\to \mathbb{R}_+$, its dual convex integrand $\delta: S^n\to \mathbb{R}_+$ is stable. (3) Let $\gamma: S^n\to \mathbb{R}_+$ be a stable convex integrand. Then, for any $i$ $(0\le i\le n)$, $\theta_0\in S^n$ is a non-degenerate critical point of $\gamma$ with Morse index $i$ if and only if $-\theta_0\in S^n$ is a non-degenerate critical point of the dual convex integrand ${\delta}$ with Morse index $(n-i)$.