We find lower bounds on the topology of the fibers F^-1 (y) ⊂ X of continuous maps F: X→ Y
in terms of combinatorial invariants of certain polyhedra and/or of the cohomology algebras …

We discuss five fundamental results of discrete mathematics: the lemmas of Sperner and
Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg …

This book is about the combinatorial properties of convex sets, families of convex sets in
finite dimensional Euclidean spaces, and finite points sets related to convexity. This area is …

In this paper we give sufficient conditions for a compactum in ℝ n to have Carathéodory
number less than n+ 1, generalizing an old result of Fenchel. Then we prove the …

The colorful Carathéodory theorem asserts that if X_1,X_2,...,X_d+1 are sets in \bfR^d, each
containing the origin 0 in its convex hull, then there exists a set S⊆X_1∪⋯∪X_d+1 with …

We study a colourful generalization of the linear programming feasibility problem, comparing
the algorithms introduced by Bárány and Onn with new methods. This is a challenging …

Given d+ 1 sets of points, or colours, S 1,…, S d+ 1 in R d, a colourful simplex is a set T⊆⋃
i= 1 d+ 1 S i such that| T∩ S i|≤ 1 for all i∈{1,…, d+ 1}. The colourful Carathéodory theorem …

A quadratic lower bound for colourful simplicial depth Page 1 J Comb Optim (2008) 16: 324–327
DOI 10.1007/s10878-008-9149-x A quadratic lower bound for colourful simplicial depth …

More Colourful Simplices Page 1 Discrete Comput Geom (2011) 45: 272–278 DOI 10.1007/s00454-010-9291-y
More Colourful Simplices Antoine Deza · Tamon Stephen · Feng Xie Received: 14 January …

The colorful Carathéodory theorem, proved by Bárány in 1982, states that given d+ 1 sets of
points S 1,…, S d+ 1 in R d, with each S i containing 0 in its convex hull, there exists a set …