**New Results at the Crossroads of Convexity, Learning and Information Theory**

I will present three new results: (i) the Cramer transform of the uniform measure on a convex body is a universal self-concordant barrier; (ii) projected Langevin Monte Carlo (i.e. discretized reflected Brownian motion with drift) allows to sample from a log-concave measure in polynomial time; and (iii) Thompson sampling combined with a multi-scale exploration solves the Bayesian convex bandit problem. The unifying theme in these results is the interplay between concepts from convex geometry, probability theory, learning and information theory.