On the second differentiability of convex surfaces
type: Item
instance of: scholarly article
title: On the second differentiability of convex surfaces
zbMATH Open document ID: 0843.26007
DOI: 10.1007/BF00150866
author: Gabriele Bianchi
author: A. Colesanti
author: Carlo Pucci
published in: Geometriae Dedicata
publication date: 1996-08-11T00:00:00Z
review text: The authors give a new proof of the known Busemann-Feller-Alexandrov theorem. This assertion points out that a real-valued convex function \(f\) on an open convex set \(\Omega\subseteq \mathbb{R}^n\) is twice differentiable almost everyhwere or equivalently that the gradient \(\nabla f\) of \(f\) is differentiable almost everyhwere. Here, the differentiability of \(\nabla f\) at the point \(x\in \Omega\) means that \(\nabla f(x)\) and a symmetric matrix \(H\) exist such that \[ y= \nabla f(x)+ H(z- x)+ o(|z- x|) \] for all \(z\in \Omega\) and all \(y\in \partial f(z)\) -- the subdifferential of \(f\) in the sense of convex analysis. In the original paper of Busemann and Feller (1935) the theorem was proved for functions with two variables, which was generalized by Alexandrov (1939) for the \(n\)-dimensional case. Regarding other proofs the authors refer to papers of Rešetnyak (1968), Bangert (1979), Rockafellar (1985), Krylov (1987), Evans/Gariepy (1992), Crandall/Ishii/Lions (1992) in which the results are proved mainly with techniques of functional analysis and distribution theory. The proof given in the present paper is based essentially on geometrical arguments. In this manner it is closely connected to the original proof of Busemann and Feller. The authors regard the second difference quotient \[ w_{x, h}(y):= {f(x+ h(y- x))- f(x)- h\nabla f(x) (y- x)\over h^2} \] and state the uniform convergence (regarding \(y\)) for \(h\) tending to zero for almost every fixed \(x\). In the main part of the proof it is shown that the limit function \(w_x(y)\) is a symmetric quadratic convex function a.e. which appears to be the second-order-term in the Taylor expansion and hence the second differential of \(f\).
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 52A41
Mathematics Subject Classification ID: 26B25
zbMATH DE Number: 856607
zbMATH Keywords: second differentiability
zbMATH Keywords: differentiability of the gradient
zbMATH Keywords: convex function
zbMATH Keywords: subdifferential
reviewed by: Jörg Thierfelder
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00150866
OpenAlex ID: W2049484389
description: scientific article; zbMATH DE number 856607