Finiteness Principles for Smooth Convex Functions

2024/02/26 by Drake, Marjorie K.
#46E35 (Primary) 26B25 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.16232

Abstract

Let E ⊂ ℝⁿ be a compact set, and f:E → ℝ. How can we tell if there exists a convex extension F ∈ C^1,1(ℝⁿ) of f, i.e. satisfying F|_E = f|_E? Assuming such an extension exists, how small can one take the Lipschitz constant Lip(∇ F): = sup_{x,y ∈ ℝⁿ, x ≠ y} (|∇ F(x) - ∇ F(y)|)/(|x-y|)? We provide an answer to these questions for the class of strongly convex functions by proving that there exist constants k^# ∈ ℕ and C>0 depending only on the dimension n, such that if for every subset S ⊂ E, #S ≤ k^#, there exists an η-strongly convex function F^S ∈ C^1,1(ℝⁿ) satisfying F^S|_S=f|_S and Lip(∇ F^S) ≤ M, then there exists an \fracηC-strongly convex function F ∈ C^1,1_c(ℝⁿ) satisfying F|_E = f|_E, and Lip(∇ F) ≤ C M²/η. Further, we prove a Finiteness Principle for the space of convex functions in C^1,1(ℝ) and that the sharp finiteness constant for this space is k^#=5.

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