Mathematics PhD defense titled "On the Differentiability Properties of Convex Functions and Convex Bodies".

Abstract: There are three main results in this thesis. We first present a new proof of a theorem that says a convex body K has boundary of class C^{1,1} if and only if there is R>0 such that K is the union of closed balls with radius R, contained in K. The first main result extends the above result to a similar characterization of C^{1,\alpha} convex bodies. Using this characterization, we find new proofs of the Kirchheim-Kristensen theorem about the differentiability of the convex envelope and the Krantz-Parks theorem about the regularity of the Minkowski sum of convex bodies. Namely we show that if a convex function f is C^{1,alpha}_{loc} and f(x) goes to infinity as |x| goes to infinity, then the convex envelope of f is C^{1,\alpha}_{loc}.  We also prove that the Minkowski sum of a convex body and a convex body of class C^{1,\alpha} is a convex body of class C^{1,\alpha}. The tools from the characterization of C^{1,1} convex bodies are used to prove the second main result, which is a new geometrically inspired proof of the Alexandrov theorem about the second order differentiability of convex functions. Moreover, we give a new proof of a result by Azagra-Hajlasz concerning the Lusin Approximation by C^{1,1} convex functions. In the third main result, we prove the set of normal directions to the k-dimensional faces on the boundary of an n-dimensional convex body is countably (n-k-1)-rectifiable. Finally we conclude by presenting characterizations of C^{1,1} and C^{1,\alpha} functions.

Advisor: Dr. Piotr Hajlasz