On the second differentiability of convex surfaces (Q1909588)

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 \(f\).