Measure-theoretic aspects of Convex bodies

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The object of the proposed research is the study of the distribution of mass of high dimensional convex bodies or more general of log-concave probability measures. Recently, several classical results from probability theory have been extended to the broader setting of these measures. These new results appear to be out of reach of the classical probabilistic reasoning based on independence which is replaced by the geometric notion of convexity. The PI intends to further investigate the geometric parameters of high-dimensional measures and in particular the geometry of generalized centroid bodies associated to these measures. With this approach and applying techniques from local theory of Banach spaces, classical convexity, probability and information theory, the PI wishes to attack various open problems related to log-concave measures such as small ball probability estimates and the hyperplane conjecture, as well as, various conjectures in the theory of convex bodies such as the regularity of the entropy numbers and the optimal bound of the minimal mean width. There is a general principle that underlies the research in this proposal: the tendency of high dimensional systems to congregate around typical forms. This is a central fact that influences the study of complex systems which appear in probability, combinatorics, statistical physics and complexity. The unexpected regularity of high dimensional convex bodies when viewed as probability spaces is expected to add a new component to our understanding of this general principle. Applications to random polytopes, random matrices, and random algorithms (topics from combinatorics, mathematical physics and informatics respectively) have already been discovered, indicating that more should be expected in the future.