Title: 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
$\gamma$ 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 $(n-i)$.