Melbourne, James2015-11-062015-11-062015-07https://hdl.handle.net/11299/175262University of Minnesota Ph.D. dissertation. July 2015. Major: Mathematics. Advisor: Sergey Bobkov. 1 computer file (PDF); vi, 59 pages.Convex measures represent a class of measures that satisfy a variant of the classical Brunn-Minkowski Inequality. Background on the associated functional and geometric inequalities is given, and the elementary theory of such measures is explored. A generalization of the Lovasz and Simonovits localization technique is developed, and some applications to large deviations are explained. In a more geometric direction, a modified Brunn-Minkowski Inequality is explored on some discrete spaces. The significance of such a notion is in its potential to serve as a definition for a lower Ricci curvature bound in non-smooth spaces.enCurvatureDiscreteLocalizationProbabilityThesis or Dissertation