Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.

Abstract. We show an equivalence between a conjecture of Bisztriczky and Fejes Tóth about arrangements 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 Tóth 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 Erdő-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erdős-Szekeres theorem of Pór and Valtr to arrangements of non-crossing convex bodies.

ABSTRACT. In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies have at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies have at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Tóth, and generalizes a theorem of Bisztriczky and Fejes Tóth. Our results on families of convex bodies are consequences of more general Ramsey-type results about the crossing patterns of systems of graphs of continuous functions f : [0, 1] → . On our way towards proving the Pach-Tóth conjecture we obtain a combinatorial characterization of such systems of graphs in which all subsystems of equal size induce equivalent crossing patterns. These highly organized structures are what we call regular systems of paths and they are natural generalizations of the notions of cups and caps from the famous theorem of Erdős and Szekeres. The characterization of regular systems is combinatorial and introduces some auxiliary structures which may be of independent interest.

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This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another set parameterizing the projective moduli space of a combinatorial polytope