Conjugacy in Convex Analysis

Abstract

In classical real analysis, the gradient of a differentiable function f : ℝⁿ → ℝ. plays a key role - to say the least. Considering this gradient as a mapping x ↦ s(x) = ∇f(x) from (some subset X of) ℝⁿ to (some subset S of) ℝⁿ, an interesting object is then its inverse: to a given s ∈ S, associate the x ∈ X such that s = ∇f(x). This question may be meaningless: not all mappings are invertible! but could for example be considered locally, taking for X x S a neighborhood of some (x ₀, s ₀ = ∇f(x ₀)), with ∇² f continuous and invertible at x ₀ (use the local inverse theorem).