Colorful linear programming, Nash equilibrium , and pivots

The colorful Carathéodory theorem, proved by Barany in 1982, states that given d+1 sets of points S_1,...,S_{d+1} in R^d, such that each S_i contains 0 in its convex hull, there exists a set subset T in the union of the S_i containing 0 in its convex hull and such that T intersects each S_i at most once. An intriguing question - still open - is whether such a set T, whose existence is ensured, can be found in polynomial time. In 1997, Barany and Onn defined colorful linear programming as algorithmic questions related to the colorful Carathéodory theorem. The question we just mentioned comes under colorful linear programming, and there are also other problems. We present new complexity results for colorful linear programming problems and propose a variant of the "Barany-Onn" algorithm, which is an algorithm computing a set T whose existence is ensured by the colorful Carathéodory theorem. Our algorithm makes a clear connection with the simplex algorithm. Some combinatorial applications of the colorful Carathéodory theorem are also discussed from an algorithmic point of view. Finally, we show that computing a Nash equilibrium in a bimatrix game is polynomially reducible to a colorful linear programming problem. On our track, we found a new way to prove that a complementarity problem belongs to the PPAD class with the help of Sperner's lemma

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

THE RIEMANN-CARATHÉODORY RESULT IMPLIES BORSUK'S RETRACT THEOREM

International audienceWe show that the Riemann-Carathéodory theorem on continuous extensions of conformal maps easily implies Borsuk's theorem characterizing those compacta K in the plane that are retracts

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