The geometry of convex domains in Euclidean space, which has its roots in the works of Minkowski and Blaschke, nowadays plays a central role in several branches of both pure and applied mathematics: functional and harmonic analysis, differential and integral geometry, the theory of partial differential equations, calculus of variations and, increasingly, in the study of certain algorithms in computer science.

Among recent striking developments in the interplay between asymptotic and classical convex geometric theories are results on volume distribution in convex bodies, including a central limit theorem for convex bodies, and logarithmic-type Sobolev inequalities with respect to log- concave probability measures showing a strong geometric-probabilistic flavor.

These results also uncovered a close link of convex geometric inequalities and the theory of optimal transport. The concentration of measure phenomenon is, in fact, an isomorphic form of isoperimetric problems, which lie at the very core of classical convex geometry. This analytical and measure theoretic approach has led to the discovery of several functional versions for many geometric inequalities and subsequently to solutions of some central problems of geometry and analysis.

At the heart of a different line of research in convex geometry lies the fundamental notion of valuation. As a generalization of the notion of measure, valuations on convex sets have long played a central role in geometry. The breakthrough in the structure theory of (translation-invariant) valuations has led to immense progress in modern integral geometry and at the same time uncovered deep connections to the theory of affine isoperimetric and analytic inequalities.

The purpose of this workshop is to focus on the latest advances and open challenges by bringing together experienced and young researchers from both the analytic and the more discrete areas of the subject who have contributed to the important recent developments in modern convex geometry.