On the Exact Constant in the Quantitative Steinitz Theorem in the Plane.
On the existence of wavelets for non-expansive dilation matrices.
Sets which simultaneously tile Rn by applying powers of an invertible matrix and translations by a lattice are studied. Diagonal matrices A for which there exist sets that tile by powers of A and by integer translations are characterized. A sufficient condition and a necessary condition on the dilations and translations for the existence of such sets are also given. These conditions depend in an essential way on the interplay between the eigenvectors of the dilation matrix and the translation lattice...
On the Expected Number of k-Sets.
On the Face-Vector of a 5-Valent Convex 3-Polytope
On the face-vectors of trivalent convex polyhedra
On the Fundamental Group of self-affine plane Tiles
Let be an expanding matrix, a set with elements and define via the set equation . If the two-dimensional Lebesgue measure of is positive we call a self-affine plane tile. In the present paper we are concerned with topological properties of . We show that the fundamental group of is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of . Furthermore, we give a short proof of the fact that the closure of each component of is a locally...
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 labellings arising from phylogenetics
We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.
On the Graph of Large Distances.
On the Graver complexity of codimension 2 matrices.
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 interpoint distance sum inequality.
On the isoperimetric-type problems with current hyperplanes.