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

Abstract

In this paper, we investigate simultaneous properties of a convex integrand $γ$ and its dual $δ$. The main results are the following three.

(1) For a $C^\infty$ convex integrand $\gamma: S^n \to \mathbf{R}_+$, its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex integrand.

(2) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Then, $γ$ is stable if and only if its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is stable.

(3) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Suppose that $γ$ 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 $γ$ 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 $δ$ with Morse index ($n-i$).