Javier Bracho

The combinatorics of colored triangulations of manifolds

Foundations for the topic of crystallizations are proposed through the more general concept of colored triangulations. Classic results and techniques of crystallizations are reviewed from this point of view. A new set of combinatorial invariants of manifolds is defined, and related to the fundamental group and other known invariants. A universal group theoretic approach for this theory is introduced.

On the minimum size of tight hypergraphs

A k-graph, H = (V, E), is tight if for every surjective mapping f: V {1,….k} there exists an edge α ϵ E sicj tjat f|α is injective. Clearly, 2-graphs are tight if and only if they are connected. Bounds for the minimum number ϕ of edges in a tight k-graph with n vertices are given. We conjecture that ϕ for every n and prove the equality when 2n + 1 is prime. From the examples, minimal embeddings of complete graphs into surfaces follow.

Very Colorful Theorems

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger’s theorem gives a generalization of the theorem of Tverberg on non-separated partitions.

Regular projective polyhedra with planar faces II

Summary. This is the first of two papers in which we classify the regular projective polyhedra in

Separoids, Their Categories and a Hadwiger-Type Theorem for Transversals

\Bbb P^3

CAROUSELS, ZINDLER CURVES AND THE FLOATING BODY PROBLEM

with planar faces. Here, we develop the basic notions; we introduce a new diophantine trigonometric equation, which plays a key role in the classification theorem, relating the combinatorial and geometric parameters of such polyhedra, and conclude with the case in which the polyhedron is an embedded surface.

The topology of the space of transversals through the space of configurations

In this paper we study the topology of transversals to a family of convex sets as a subset of a Grassmanian manifold. This topology seems to be ruled by a combinatorial structure which we call a separoid. With these combinatorial objects and the topological notion of virtual transversal we prove a Borsuk—Ulam-type theorem which has as a corollary a generalization of Hadwiger’s theorem.

A colorful theorem on transversal lines to plane convex sets

A carousel is a dynamical system that describes the movement of an equilateral linkage in which the midpoint of each rod travels parallel to it. They are closely related to the floating body problem. We prove, using the work of Auerbach, that any figure that floats in equilibrium in every position is drawn by a carousel. Of special interest are such figures with rational perimetral density of the floating chords, which are then drawn by carousels. In particular, we prove that for some perimetral densities the only such figure is the circle, as the problem suggests.

A Classification Theorem for Zindler Carrousels

Abstract Let F be a family of convex sets in R n and let Tm(F) be the space of m-transversals to F as subspace of the Grassmannian manifold. The purpose of this paper is to study the topology of Tm(F) through the polyhedron of configurations of (r+1) points in R n . This configuration space has a natural polyhedral structure with faces corresponding to what has been called order types. In particular, if r=m+1 and Tm−1(F) is nonempty, we prove that the homotopy type of Tm(F) is ruled by the set of all possible order types achieved by the m-transversals of F. We shall also prove that the set of all m-transversals that intersect F with a prescribed order type is a contractible space.

Immobilization of smooth convex figures

We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line.