On the Second Differentiability of Convex Functions

Abstract: In 1939 Alexandrov proved one of the most beautiful results in convex analysis concerning the second order differentiability of a convex function. The result is similar to Rademacher's theorem and states that a convex function is second differentiable almost everywhere. Since the original publication, numerous proofs have been given for this famed theorem and in this talk we will look at the most recent proof by Azagra, Cappello, and Hajlasz. The proof is elementary and all material not covered in a standard advanced calculus or measure theory class will be explained carefully.