Convexity is a simple idea that manifests itself in a surprising variety of places. This fertile
field has an immensely rich structure and numerous applications. Barvinok demonstrates …

This paper studies fault-tolerant algorithms for the problem of gathering N autonomous
mobile robots. A gathering algorithm, executed independently by each robot, must ensure …

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 …

The (p, q) theorem of Alon and Kleitman asserts that if F is a family of convex sets in R d
satisfying the (p, q) condition for some p⩾ q⩾ d+ 1 (ie among any p sets of F, some q have a …

The linear response of two-dimensional amorphous elastic bodies to an external delta force
is determined in analogy with recent experiments on granular aggregates. For the generated …

The concept of location depth was introduced as a way to extend the univariate notion of
ranking to a bivariate configuration of data points. It has been used successfully for robust …

Let (P, E) be a (d+ 1)-uniform geometric hypergraph, where P is an n-point set in general
position in ℝ d and is a collection of d-dimensional simplices with vertices in P, for 0< ɛ≤ 1 …

Given a finite point set P in ℝ d, and> 0 we say that N⊆ ℝ d is a weak-net if it pierces every
convex set K with| K∩ P|≥ є| P|. Let d≥ 3. We show that for any finite point set in ℝ d, and …

We show that for any finite point set P in the plane and ε> 0 there exist O (1/ε 3/2+ γ) points,
for arbitrary small γ> 0, that pierce every convex set K with| K∩ P|≥ ε| P|. This is the first …

We show that very weak topological assumptions are enough to ensure the existence of a
Helly-type theorem. More precisely, we show that for any non-negative integers b and d …