In this paper we present a variety of problems in the interface between combinatorics and
geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these …

In this paper we settle a special case of the Grassmann convexity conjecture formulated
earlier by B. and M. Shapiro. We present a conjectural formula for the maximal total number …

Let $ T (d, r)=(r-1)(d+ 1)+ 1$ be the parameter in Tverberg's theorem, and call a partition
$\mathcal I $ of $\{1, 2,\ldots, T (d, r)\} $ into $ r $ parts a" Tverberg type". We say that …

We continue a sequence of recent works studying Ramsey functions for semialgebraic
predicates in Rd. A k-ary semialgebraic predicate Φ (x1,…, xk) on Rd is a Boolean …

The tolerant Tverberg theorem generalizes Tverberg's theorem by introducing a new
parameter t called tolerance. It states that there is a minimal number N so that any set of at …

For positive integers N and r≥ 2, an r-monotone coloring of ({1,…, N} r) is a 2-coloring by− 1
and+ 1 that is monotone on the lexicographically ordered sequence of r-tuples of every (r+ …

Given a finite point set P ⊂ R^ d, we call a multiset A a one-sided weak ɛ-approximant for P
(with respect to convex sets), if | P ∩ C |/| P |-| A ∩ C |/| A | ≦ ε for every convex set C. We …

The intersection graph of a set system S is a graph on the vertex set S, in which two vertices
are connected by an edge if and only if the corresponding sets have nonempty intersection …

In this paper we give an asymptotically tight bound for the tolerated Tverberg Theorem when
the dimension and the size of the partition are fixed. To achieve this we study certain …

Saldanha, Nicolau ; Shapiro, Boris ; Shapiro, Michael Grassmann convexity and multiplicative
Sturm theory. Revisited. (English) Page 1 Saldanha, Nicolau; Shapiro, Boris; Shapiro, Michael …