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

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 $(n-i)$.