Luis Montejano

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.

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.

Characterization of n-path graphs and of graphs having nth root

Abstract We characterize connected graphs and digraphs having an n th root and so generalize results by A. Mukhopadhyay and D. P. Geller, respectively. We then define the n -path graph of a graph and characterize those graphs which are n -path graphs. This extends recent results by B. Devadas Acharya and M. N. Vartak. The corresponding problem for digraphs is also considered.

A quick proof of Singhof's cat(M ×S1)=cat(M)+1 theorem

For a topological space X, the Lusternik-Schnirelmann category, cat(X), is the smallest number N such that X can be covered by N open subsets each of which is contractible in X. W. Singhof [6] proved that the minimal number of n-balls which suffice to cover a closed PL n-manifold M coincides with the Lusternik-Schnirelmann category if the latter is not too small compared with the dimension of M. As a consequence, one obtains in this case that cat(M×1)=cat(M)+1, thus establishing a special case of a long-standing conjecture. The purpose of this paper is to give a quick proof of Singhof s results by exploiting the linear structure between the k-skeleton of a polyhedron and its dual skeleton.

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

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.

Order Types of Convex Bodies

We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turns out to be an oriented matroid. We also give new upper bounds on the Erdős–Szekeres theorem in the context of convex bodies.

CAROUSELS, ZINDLER CURVES AND THE FLOATING BODY PROBLEM

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.

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

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.

Tolerance in Helly-Type Theorems

In this paper we introduce the notion of tolerance in connection with Helly-type theorems and prove, using the Erdős–Gallai theorem, that any Helly-type theorem can be generalized by relaxing the assumptions and conclusion, allowing a bounded number of exceptional sets or points. In particular, we analyze some of the classical Helly-type theorems, such as Caratheodory’s and Tverberg’s theorems, as well as some other interesting ones.

A New Topological Helly Theorem and Some Transversal Results

We prove that for a topological space