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

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 $C^{\mathrm{\infty }}$ convex integrand $\gamma :S^n\to {\mathbb{R}}_{+}$, its dual convex integrand $\delta :S^n\to {\mathbb{R}}_{+}$ is of class $C^{\mathrm{\infty }}$ if and only if $\gamma$ is a strictly convex integrand. (2) Let $\gamma :S^n\to {\mathbb{R}}_{+}$ be a $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ 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 $(n-i)$.