Abstract:

A classical result (Alexandroff-Buselman-Feller theorem) is that convex functions are punctually second-order differentiable almost everywhere. The paper "An Alexsandrov type theorem for k-convex functions" by Nirmalendu Chaudhuri and Neil Trudinger extends the result to all k-convex functions for k>n/2. Here the k-convexity is understood in the viscosity sense. Under stronger assumptions on the function, the claim holds for arbitrary n and k. This result was used by Ravi Shanker and Yuan Yu in their work "Hessian estimates for the sigma-2 equation in dimension four" which they gave a seminar talk here last semester. Recently, we have been studying the regularity of sigma k Schouten equation in the negative cone and found this result useful. We would like to go through the details in this seminar.