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

In this paper, the following three are shown. (1) For a $C^{\mathrm{\infty }}$ convex integrand $\gamma :S^n\to {\mathbb{R}}_{+}$, its dual convex integrand $\delta: S^n\to $\mathbb{R}$+$isofclass$C^\infty$\mathrm{.}(2)Forastableconvexintegrand$\gamma: S^n\to $\mathbb{R}$+$,itsdualconvexintegrand$\delta: S^n\to $\mathbb{R}$+$isstable\mathrm{.}(3)Let$\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 $(n-i)$.