On the Busemann-Petty Problem for Perturbations of the Ball.
On the centre of gravity and width of lattice-constrained convex sets in the plane.
On the characterization of some families of closed convex sets.
On the curves of constant relative width
On the Erdös-Debrunner inequality.
On the hessian of the optimal transport potential
We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse...
On the isoperimetric-type problems with current hyperplanes.
On the isoperimetry of graphs with many ends
Let X be a connected graph with uniformly bounded degree. We show that if there is a radius r such that, by removing from X any ball of radius r, we get at least three unbounded connected components, then X satisfies a strong isoperimetric inequality. In particular, the non-reduced -cohomology of X coincides with the reduced -cohomology of X and is of uncountable dimension. (Those facts are well known when X is the Cayley graph of a finitely generated group with infinitely many ends.)
On the isotropic constant of marginals
We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in , i ≤ m, then for every F in the Grassmannian , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.
On the length of some curves in the unit sphere
On the Maximum of the Minimum of a Sum.
On the Minimal Convex Annulus of a Planar Convex Body.
On the radius of a set in a Hilbert space
On the section of a lattice–covering of balls
On the symmetric difference metric for convex bodies.
On the Union of Jordan Regions and Collision-Free Translational Motion Amidst Polygonal Obstacles.
On the volume of a certain polytope.
On the Volume of Equichordal Sets.
On the volume of the double stochastic matrices.
On the ψ₂-behaviour of linear functionals on isotropic convex bodies
The slicing problem can be reduced to the study of isotropic convex bodies K with , where is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that for all θ in a subset U of with measure σ(U) ≥ 1 - exp(-c√n). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that . In a different direction, we show that good average ψ₂-behaviour of linear functionals on an isotropic...