Li, Dongbin2023-05-162023-05-162023https://udspace.udel.edu/handle/19716/32747The main theme of this dissertation is to explore four topics at the intersection of information theory, probability, discrete/convex geometry and geometric functional analysis. ☐ In the first part of the dissertation, we introduce an information-theoretic approach to the Kneser-Poulsen conjecture in discrete geometry first formulated by Poulsen in 1954 and Kneser in 1955. Our approach revolves around a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a 1-Lipschitz map. We answer this broad question affirmatively in various cases. ☐ In the second part, we characterize the convex potentials, which is also known as the information content, of log-concave densities in ℝn under various assumptions. Built upon this characterization, we show that, for a subclass of log-concave densities, the normalized information content satisfies both a central limit theorem and a large deviation principle, which, on one hand, generalizes the result for Gaussian random vectors, on the other hand, sheds some light on a more general conjecture. ☐ In the third part, we investigate a metric generalization of Rényi entropies that are called diversities. We show that on the Euclidean metric space ℝn, one may recover the Rényi entropies from diversities. And via diversities, we define the diversity dimension of different orders of a Borel probability measure, and prove that they coincide with the information dimensions defined via Rényi entropies. Meanwhile, the fact that maximum diversity can also be used to recover some geometric invariants motivates us to generalize some classic sumset inequalities with maximum diversity now playing the role of "size". The relationship with the other notion of diversity is also explored in this section.Discrete/convex geometryInformation theoryProbabilityThesis1379190357https://doi.org/10.58088/88xk-8t692023-03-22en