Topology of geometric joins

We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carath\'eodory and Tverberg theorems, and their relatives. We conjecture that when the family has at least $d+1$ sets, where $d$ is the dimension of the space, then the geometric join is contractible. We are able to prove this when $d$ equals $2$ and $3$, while for larger $d$ we show that the geometric join is contractible provided the number of sets is quadratic in $d$. We also consider a matroid generalization of geometric joins and provide similar bounds in this case

Optimal bounds for the colored Tverberg problem

We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Barany et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Barany & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.Comment: 17 pages, 3 figures; revised version (February 2013), to appear in J. European Math. Soc. (JEMS

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

A Theorem of Barany Revisited and Extended

International audienceThe colorful Caratheodory theorem states that given d+1 sets of points in R^d, the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the colorful Caratheodory theorem: given d/2+1 sets of points in $R^d, and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a (d/2+1)-dimensional rainbow simplex intersecting C

An Optimal Generalization of the Colorful Carathéodory Theorem

International audienceThe Colorful Carathéodory theorem by Bárány (1982) states that given d + 1 sets of points in R d , the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d + 1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the Colorful Carathéodory theorem: given + 1 sets of points in R d and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a rainbow simplex intersecting C