A geometric approach to second-order differentiability of convex functions

Author:

Azagra Daniel,Cappello Anthony,Hajłasz Piotr

Abstract

We show a new, elementary and geometric proof of the classical Alexandrov theorem about the second order differentiability of convex functions. We also show new proofs of recent results about Lusin approximation of convex functions and convex bodies by C 1 , 1 C^{1,1} convex functions and convex bodies.

Funder

National Science Foundation

Publisher

American Mathematical Society (AMS)

Subject

Geometry and Topology,Discrete Mathematics and Combinatorics,Analysis,Algebra and Number Theory

Reference16 articles.

1. Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it;Alexandroff, A. D.;Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser.,1939

2. Global and fine approximation of convex functions;Azagra, Daniel;Proc. Lond. Math. Soc. (3),2013

3. Locally 𝐶^{1,1} convex extensions of 1-jets;Azagra, Daniel;Rev. Mat. Iberoam.,2022

4. Daniel Azagra, Marjorie Drake, and Piotr Hajłasz, 𝐶²-Lusin approximation of strongly convex functions, Preprint, 2023.

5. Lusin-type properties of convex functions and convex bodies;Azagra, Daniel;J. Geom. Anal.,2021

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