Singular Sets of Generalized Convex Functions

In the first part of the dissertation we prove that, under quite general conditions on a cost function c in RR^n, the Hausdorff dimension of the singular set of a c-concave function has dimension at most n-1. Our result applies for non-semiconcave cost functions and has applications in optimal mass transportation. The purpose of the second part of the thesis is to extend a result of Alberti and Ambrosio about singularity sets of monotone multivalued maps to the sub-Riemannian setting of Heisenberg groups. We prove that the k-th horizontal singular set of a H-monotone multivalued map of the Heisenberg group HH^n, with values in RR^{2n}, has Hausdorff dimension at most 2n+2-k.