The aim of this talk is to report on some modern aspects of classical Convex Geometry, in particular, of the Brunn-Minkowski Theory. The theory of Convex Bodies (convex and compact subsets of $\mathbb R^n$) in which the Brunn-Minkowski Theory is originally framed on, is a classical central theme in metric geometry. Convex Geometry, as the geometry of convex domains in the Euclidean space, has inherent geometric and analytic connections, as well as further links with other fields within Mathematics, and beyond. The Brunn-Minkowski theory is based on the combination of the Minkowski sum of convex sets with the notion of volume (Lebesgue measure). Classical notions in this context are mixed volumes, in particular, quermassintegrals; geometric inequalities, especially the BrunnMinkowski inequality, as well as the equality cases of them; and polynomial expansions, as the Steiner formula.

In the last two decades, the classical theory of convex bodies has been fruitfully enriched, expanded, and generalized. The replacement of the Minkowski sum by other combinations of convex bodies (or more general sets), the interplay of convex sets and functions, or the theory of valuations on the space of convex bodies are some examples of these successful and far-reaching new aspects of Convex Geometry.

Within the classical theory, there are still several open questions. Many of these are concerned with equality cases of inequalities on convex bodies, and improvements of these inequalities. Some others ask for geometric aspects of the structure of convex bodies. In this talk, we will concentrate on some aspects of the classical theory of convex bodies for which there are still some questions to answer. For each of them, a short introduction providing the necessary background and the state of the art of questions (and answers) will be provided. In particular, we will deal with various aspects of the Steiner polynomial and the Brunn-Minkowski inequality. We will also discuss some decomposition aspects of convex bodies by means of Minkowski sums of related convex bodies. In the last part, we will deal with certain special families of convex bodies associated with a given convex body, and the behavior of some functions restricted to these families.

Contact Jacob Schach Møller jacob@math.au.dk, for Zoom details.