2020 Volume 43 Issue 2 Pages 221-242
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: Sn → R+, its dual convex integrand δ: Sn → R+ is of class C∞ if and only if γ is a strictly convex integrand.
(2) Let γ: Sn → R+ be a C∞ strictly convex integrand. Then, γ is stable if and only if its dual convex integrand δ: Sn → R+ is stable.
(3) Let γ: Sn → R+ be a C∞ strictly convex integrand. Suppose that γ is stable. Then, for any i (0 ≤ i ≤ n), a point θ0 ∈ Sn is a non-degenerate critical point of γ with Morse index i if and only if its antipodal point -θ0 ∈Sn is a non-degenerate critical point of the dual convex integrand δ with Morse index (n-i).
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