Simultaneous smoothness and simultaneous stability of a $C^\infty$ strictly convex integrand and its dual

TL;DR

This paper explores the smoothness and stability properties of a smooth convex integrand and its dual, establishing conditions for their equivalence and describing the relationship between their critical points and Morse indices.

Contribution

It provides new conditions linking the smoothness and stability of convex integrands with their duals, including the Morse index relationship of their critical points.

Findings

Dual integrand is smooth iff the original is strictly convex.

Stability of the integrand is equivalent to the stability of its dual.

Critical points of the integrand correspond to antipodal critical points of the dual with complementary Morse indices.

Abstract

In this paper, we investigate simultaneous properties of a convex integrand $\gamma$ and its dual $\delta$. The main results are the following three. (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 $C^\infty$ if and only if $\gamma$ is a strictly convex integrand. (2) Let $\gamma: S^n\to \mathbb{R}_+$ be a $C^\infty$ strictly convex integrand. Then, $\gamma$ is stable if and only if its dual convex integrand $\delta: S^n\to \mathbb{R}_+$ is stable. (3) Let $\gamma: S^n\to \mathbb{R}_+$ be a $C^\infty$ strictly convex integrand. Suppose that $\gamma$ is stable. Then, for any $i$ $(0\le i\le n)$, a point $\theta_0\in S^n$ is a non-degenerate critical point of $\gamma$ with Morse index $i$ if and only if its antipodal point $-\theta_0\in S^n$ is a non-degenerate critical point of the dual convex integrand $\delta$ with Morse index $(n-i)$.