The logarithmic Minkowski inequality for non-symmetric convex bodies

Blaschke- and Minkowski-endomorphisms of Convex Bodies

We consider maps of the family of convex bodies in Euclidean ddimensional space into itself that are compatible with certain structures on this family: A Minkowski-endomorphism is a continuous, Minkowski-additive map that commutes with rotations. For d ≥ 3, a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by app...

On the Isotropic Constant of Non–Symmetric Convex Bodies

We show that Bourgain’s estimate LK ≤ cn 1 4 logn for the isotropic constant holds true for non-symmetric convex bodies as well.

A Curved Brunn Minkowski Inequality for the Symmetric Group

In this paper, we construct an injection A × B → M ×M from the product of any two nonempty subsets of the symmetric group into the square of their midpoint set, where the metric is that corresponding to the conjugacy class of transpositions. If A and B are disjoint, our construction allows to inject two copies of A × B into M ×M . These injections imply a positively curved Brunn-Minkowski inequ...

The Brunn-Minkowski-Type Inequality

Entropy and Asymptotic Geometry of Non–Symmetric Convex Bodies

We extend to the general, not necessary centrally symmetric setting a number of basic results of Local Theory which were known before for centrally symmetric bodies and were using very essentially the symmetry in their proofs. Some of these extensions look surprising. The main additional tool is a study of volume behavior around the centroid of the body.