Tymoshkevych T. Geometric properties of probability measures in linear spaces and functional inequalities

The thesis is devoted to geometric characteristics such as zonoid and lift zonoid, proof of functional inequalities, and the connection between them. Introduced generalizations of zonoid, lift zonoid and trimmed regions with maximum generality, for infinite-demensional space of the cylindrical measure and summarizes relevant functional inequality, naturally, summarizing the criteria inequalities in terms of lift zonoids. The results shown in the Poincare inequality for a class of Gaussian censorial measures set constants significantly better than those obtained with the classical criteria where we need condition on the second derivative of the density measure. Also by martingale method given estimates for entropy and variation.