Kasia Wyczesany (University of Leeds) – Dualities and Extremal Inequalities in Convex Geometry

Convex geometry has long been influenced by the study of dualities and extremal inequalities, with origins in classical affine geometry and functional analysis. In this talk, we will explore an abstract concept of duality, focusing on the classical idea of the polar set, which captures the duality of finite-dimensional normed spaces. This notion leads to fundamental questions about volume products, inspiring some of the most famous inequalities in the field. While we will mention Mahler’s influential 1939 conjecture regarding the minimiser of the volume product, the emphasis will be on the Blaschke–Santaló inequality, which identifies the maximiser, along with its modern extensions.