Simultaneous smoothness and simultaneous stability of a $C^\infty$ strictly convex integrand and its dual
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 $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex integrand. (2) Let $γ: S^n\to \mathbb{R}_+$ be a $C^\infty$ strictly convex integrand. Then, $γ$ is stable if and only if its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is stable. (3) Let $γ: S^n\to \mathbb{R}_+$ be a $C^\infty$ strictly convex integrand. Suppose that $γ$ is stable. Then, for any $i$ $(0\le i\le n)$, a point $θ_0\in S^n$ is a non-degenerate critical point of $γ$ with Morse index $i$ if and only if its antipodal point $-θ_0\in S^n$ is a non-degenerate critical point of the dual convex integrand $δ$ with Morse index $(n-i)$.