Moment Matrices and SL($n$) Equivariant Valuations on Polytopes

All SL($n$) equivariant symmetric matrix valued valuations on convex polytopes in ${\mathbb{R}}^n$ are completely classified without any continuity assumptions. The unique ones turn out to be the moment matrices corresponding to the classical Legendre ellipsoid and the isotropic position.

On the intersection of random rotations of a symmetric convex body

Let C be a symmetric convex body of volume 1 in ${\mathbb R}^n$. We provide general estimates for the volume and the radius of C ∩ U(C) where U is a random orthogonal transformation of ${\mathbb R}^n$. In particular, we consider the case where C is in the isotropic position or C is the volume normalized Lq-centroid body Zq(μ) of an isotropic log-concave measure μ on ${\mathbb R}^n$.

On the volume of caps and bounding the mean-width of an isotropic convex body

LetKbe a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting onKexhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if anarbitraryisotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width.