On the Erdös-Debrunner inequality.
On the estension of Lipschitz functions with respect to two Hilbert norms and two Lipschitz conditions
On the Euler Characteristic in Spaces with a Separability Property.
On the Euler Characteristic of Finite Unions of Convex Sets.
On the Exact Constant in the Quantitative Steinitz Theorem in the Plane.
On the Expected Number of k-Sets.
On the general Brunn-Minkowski theorem.
On the General Motion-Planning Problem with Two Degrees of Freedom.
On the geometry of convex reflectors
In this paper we consider a special class of convex hypersurfaces in Euclidean space which arise as weak solutions to some inverse problems of recovering reflectors from scattering data. For this class of hypersurfaces we study the notion of the focal function which, while sharing the important convexity property with the classical support function, has the advantage of being exactly the "right tool" for such inverse problems. We also discuss briefly the close analogy between one such inverse problem...
On the geometry of and .
On the Geometry of Stable Compact Convex Sets.
On the Graph of Large Distances.
On the Hadwiger numbers of centrally symmetric starlike disks.
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 Integral Representation in Convex Noncompact Sets of Tight Measures.
On the Intermediate Area Functions of Convex Bodies.
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