Majorization: in information theory, analysis, and beyond.

Abstract: Modulo permutations, the notion of majorization gives a useful partial order on the set of n-dimension probability vectors. Classically in probability and information theory, the R\’enyi entropy reverses this ordering, a property referred to as Schur-concavity. We will explore the notion of majorization for proving analytic inequalities. In particular we present a transportation argument that yields a majorizing relationship between densities. As applications, elementary derivations of certain Fourier theoretic L^p norm comparisons can be obtained. Time permitting, we will discuss geometric and combinatorial consequences of the obtained bounds.