Deborah Oliveros

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.

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.

Colourful and Fractional (p,q)-theorems

Let p≥q≥d+1 be positive integers and let

A Classification Theorem for Zindler Carrousels

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The intersection condition for regular polytopes

be a finite family of convex sets in

Colorful associahedra and cyclohedra

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About the piercing number of a family of intervals

. Assume that the elements of

A fractional Helly theorem for boxes

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