Simultaneous smoothness and simultaneous stability of a C^∞ strictly convex integrand and its dual

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: 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).