On the global shape of continuous convex functions on Banach spaces

2019/10/28 by Azagra, Daniel
#FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1910.12520

Abstract

We make some remarks on the global shape of continuous convex functions defined on a Banach space Z. Among other results we prove that if Z is separable then for every continuous convex function f:Z→ℝ there exist a unique closed linear subspace Y_f of Z such that, for the quotient space X_f :=Z/Y_f and the natural projection π:Z→ X_f, the function f can be written in the form f(z)=φ(π(z)) +ℓ(z) \textrm for all z∈ Z, where ℓ_f∈ X^* and φ:X_f→ℝ is a convex function such that lim_t→∞φ(x+tv)=∞ for every x, v∈ X_f with v≠ 0. This kind of result is generally false if Z is nonseparable (even in the Hilbertian case Z=ℓ₂(Γ) with Γ an uncountable set).

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