Kodai Mathematical Journal
Online ISSN : 1881-5472
Print ISSN : 0386-5991
ISSN-L : 0386-5991
Simultaneous smoothness and simultaneous stability of a C strictly convex integrand and its dual
Erica Boizan BatistaHuhe HanTakashi Nishimura
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2020 Volume 43 Issue 2 Pages 221-242

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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 convex integrand γ: SnR+, its dual convex integrand δ: SnR+ is of class C if and only if γ is a strictly convex integrand.

(2) Let γ: SnR+ be a C strictly convex integrand. Then, γ is stable if and only if its dual convex integrand δ: SnR+ is stable.

(3) Let γ: SnR+ be a C strictly convex integrand. Suppose that γ is stable. Then, for any i (0 ≤ in), a point θ0Sn is a non-degenerate critical point of γ with Morse index i if and only if its antipodal point -θ0Sn is a non-degenerate critical point of the dual convex integrand δ with Morse index (n-i).

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