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June 2020 Simultaneous smoothness and simultaneous stability of a $C^\infty$ strictly convex integrand and its dual
Erica Boizan Batista, Huhe Han, Takashi Nishimura
Kodai Math. J. 43(2): 221-242 (June 2020). DOI: 10.2996/kmj/1594313551

Abstract

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 $\gamma: S^n \to \mathbf{R}_+$, its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex integrand.

(2) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Then, $γ$ is stable if and only if its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is stable.

(3) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Suppose that $γ$ 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 $γ$ 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 $δ$ with Morse index ($n-i$).

Citation

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Erica Boizan Batista. Huhe Han. Takashi Nishimura. "Simultaneous smoothness and simultaneous stability of a $C^\infty$ strictly convex integrand and its dual." Kodai Math. J. 43 (2) 221 - 242, June 2020. https://doi.org/10.2996/kmj/1594313551

Information

Published: June 2020
First available in Project Euclid: 9 July 2020

zbMATH: 07227747
MathSciNet: MR4121360
Digital Object Identifier: 10.2996/kmj/1594313551

Rights: Copyright © 2020 Tokyo Institute of Technology, Department of Mathematics

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Vol.43 • No. 2 • June 2020
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