The classical Loomis-Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide upper bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer's dual Loomis-Whitney inequality. Joint work with S. Brazitikos and D-M. Liakopoulos. We also discuss applications of these inequalities to questions regarding the surface area of lower dimensional projections and the average section functional of lower dimensional sections of a convex body, which were the motivation for them (joint works with A. Koldobsky, S. Dann, S. Brazitikos and P. Valettas).

The aim of the talks will be to present the main results on the geometry of isotropic convex bodies in connection with a number of well-known open questions regarding the distribution of volume in high-dimensional convex bodies, such as the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture.