2025/12/15 by Thomas Deck, Deck, Thomas, Carlos Mudarra +1
Mathematics · #Advanced Banach Space Theory #Banach space #Bounded function #Classical Analysis and ODEs (math.CA) #Constant (computer programming) #Convex set #Differentiable function #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Hilbert space #Holomorphic and Operator Theory #Lipschitz continuity #Regular polygon #Simple (philosophy)
paper · pdf · doi:10.48550/arxiv.2512.13324
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2025/12/15 · openalex created_date 2025/12/17 · openalex updated_date 2026/08/06
Given a superreflexive Banach space X, and a set E ⊂ X, we characterise the 1-jets (f,G) on E that admit C1,ω convex extensions (F,DF) to all of X; where ω is any admissible modulus of continuity depending on the regularity of X. Moreover, we obtain precise estimates for the growth of the C1,ω seminorm of the extensions with respect to the initial data. We show how these estimates can be improved in the Hilbert setting, and are asymptotically sharp for Hölder moduli. Remarkably, our extensions have the sharp Lipschitz constant Lip(F,X) = ‖G‖L^∞(E), when G is a bounded map. All these extensions are given by simple and explicit formulas. We also prove a similar theorem for C1 convex extensions of jets defined on compact subsets E of superreflexive spaces X, with the sharp Lipschitz constant too. The results are new even when X=ℝn.