Michael Gene Dobbins

THE ERDŐS–SZEKERES PROBLEM FOR NON-CROSSING CONVEX SETS

We show an equivalence between a conjecture of Bisztriczky and Fejes Toth about families of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and Toth on the Erdős–Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk on the Erdős–Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erdős–Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erdős–Szekeres theorem of Por and Valtr to families of non-crossing convex bodies.

A point in a \(nd\)-polytope is the barycenter of \(n\) points in its \(d\)-faces

Using equivariant topology, we prove that it is always possible to find

Realizability of Polytopes as a Low Rank Matrix Completion Problem

n

The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions

n points in the

Realization Spaces of Arrangements of Convex Bodies

d

∀ ∃ \mathbb R ∀ ∃ R -Completeness and Area-Universality.

d-dimensional faces of a