18 research outputs found
A Theorem of Barany Revisited and Extended
International audienceThe colorful Caratheodory theorem states that given d+1 sets of points in R^d, the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the colorful Caratheodory theorem: given d/2+1 sets of points in $R^d, and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a (d/2+1)-dimensional rainbow simplex intersecting C
Tropical Carathéodory with Matroids
Bárány’s colorful generalization of Carathéodory’s Theorem combines geometrical and combinatorial constraints. Kalai–Meshulam (2005) and Holmsen (2016) generalized Bárány’s theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert–Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth
Discrete Geometry and Convexity in Honour of Imre Bárány
This special volume is contributed by the speakers of the Discrete Geometry and
Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference
is to celebrate the 70th birthday and the scientific achievements of professor
Imre Bárány, a pioneering researcher of discrete and convex geometry, topological
methods, and combinatorics. The extended abstracts presented here are written by
prominent mathematicians whose work has special connections to that of professor
Bárány. Topics that are covered include: discrete and combinatorial geometry,
convex geometry and general convexity, topological and combinatorial methods.
The research papers are presented here in two sections. After this preface and a
short overview of Imre Bárány’s works, the main part consists of 20 short but very
high level surveys and/or original results (at least an extended abstract of them)
by the invited speakers. Then in the second part there are 13 short summaries of
further contributed talks.
We would like to dedicate this volume to Imre, our great teacher, inspiring
colleague, and warm-hearted friend
Radon Numbers Grow Linearly
Define the k-th Radon number r_k of a convexity space as the smallest number (if it exists) for which any set of r_k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that r_k grows linearly, i.e., r_k ? c(r?)? k
Matching points with disks with a common intersection
We consider matchings with diametral disks between two sets of points R and
B. More precisely, for each pair of matched points p in R and q in B, we
consider the disk through p and q with the smallest diameter. We prove that for
any R and B such that |R|=|B|, there exists a perfect matching such that the
diametral disks of the matched point pairs have a common intersection. In fact,
our result is stronger, and shows that a maximum weight perfect matching has
this property
Bounding Helly Numbers via Betti Numbers
We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b,d) such that the following holds. If F is a finite family of subsets of R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)/2, then F has Helly number at most h(b,d). These topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent interest
Contaje de triángulos en conjuntos de puntos coloreados: un problema de la geometría combinatoria
A classical object of study in combinatorial geometry are sets S of points in the plane. A triangle with vertices from S is called empty if it contains no points of S in its interior. The number of empty triangles depends on the positions of points from S and a burning question is: How many empty triangles are there at least, among all sets S of n points? In order to discard degenerate point configurations, we only consider sets S without three collinear points. In this project, a software has been developed which allows to count the number of empty triangles in a set of n points in the plane. The software permits generation of point sets and their graphical visualization, as well as searching and displaying of optimal point configurations encountered. A point set of a given cardinality is said to be optimal if it contains the minimum number of empty triangles. The objective is to derive bounds on the minimum number of empty triangles by means of experiments realized with our software. The created program also allows to count empty monochromatic triangles in two-colored point sets. A triangle is called monochromatic if its three vertices have the same color. While the first problem has been studied extensively during the last decades, the two-colored version remains to be explored in depth. In this work we also expose our results on the minimum number of empty triangles in (small) two-colored point sets. Also, the treated problem is put in context with related results, such as the Erdös-Szekeres theorem, and a short outline of famous problems which contributed to the rise of combinatorial geometry is presented.Un objeto clásico de estudio en la Geometría combinatoria son conjuntos S de n puntos en el plano. Se dice que un triángulo con vértices en S esta vacío si no contiene puntos de S en su interior. El número de triángulos vacíos depende de cómo se dibujó el conjunto S y una pregunta ardiente es:
¿Cuántos triángulos vacíos hay como mínimo en cada conjunto S de n puntos? Para descartar configuraciones de puntos degeneradas solo se consideran nubes de puntos sin tres puntos colineales
The convexification effect of Minkowski summation
Let us define for a compact set the sequence It was independently proved by Shapley, Folkman and Starr (1969)
and by Emerson and Greenleaf (1969) that approaches the convex hull of
in the Hausdorff distance induced by the Euclidean norm as goes to
. We explore in this survey how exactly approaches the convex
hull of , and more generally, how a Minkowski sum of possibly different
compact sets approaches convexity, as measured by various indices of
non-convexity. The non-convexity indices considered include the Hausdorff
distance induced by any norm on , the volume deficit (the
difference of volumes), a non-convexity index introduced by Schneider (1975),
and the effective standard deviation or inner radius. After first clarifying
the interrelationships between these various indices of non-convexity, which
were previously either unknown or scattered in the literature, we show that the
volume deficit of does not monotonically decrease to 0 in dimension 12
or above, thus falsifying a conjecture of Bobkov et al. (2011), even though
their conjecture is proved to be true in dimension 1 and for certain sets
with special structure. On the other hand, Schneider's index possesses a strong
monotonicity property along the sequence , and both the Hausdorff
distance and effective standard deviation are eventually monotone (once
exceeds ). Along the way, we obtain new inequalities for the volume of the
Minkowski sum of compact sets, falsify a conjecture of Dyn and Farkhi (2004),
demonstrate applications of our results to combinatorial discrepancy theory,
and suggest some questions worthy of further investigation.Comment: 60 pages, 7 figures. v2: Title changed. v3: Added Section 7.2
resolving Dyn-Farkhi conjectur