Functional Inequalities and Concentration of Measure I

Explore concentration inequalities, a fundamental tool in probability and asymptotic geometric analysis, in this 47-minute lecture by Radek Adamczak. Delve into the softer approach to concentration based on functional inequalities, focusing on classical inequalities like the Poincaré inequality and log-Sobolev inequality. Examine their common properties, such as tensorization, and discover how they lead to various forms of concentration for Lipschitz functions. Begin with the continuous setting, starting with exponential and Gaussian measures, before moving on to discrete examples. Learn about concentration results for non-Lipschitz functions derived from functional inequalities. Cover topics including Gaussian concentration inequality, variance and entropy, variational definitions, the lobster of inequality, Gaussian measure, probabilistic proofs, and Herb's argument.

Introduction
Gaussian concentration inequality
Variance and entropy
Variational definitions
Tensorization properties
Functional inequalities
Lobster of inequality
Gaussian measure
Probabilistic proof
Herbs argument
Summary
Proof