arXiv subjects

Pablo Soberón

Publications and source records attributed to Pablo Soberón.

At least 55 records · Page 3Linked to original sources

arXiv2020

Quantitative combinatorial geometry for concave functions

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colorful and fractional variants of all our Helly-type theorems.

math.CO↗

arXiv2020

Non-commutative groups as prescribed polytopal symmetries

We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.

math.MG↗

arXiv2019

Balanced convex partitions of lines in the plane

We prove an extension of a ham sandwich theorem for families of lines in the plane by Dujmović and Langerman. Given two sets $A, B$ of $n$ lines each in the plane, we prove that it is possible to partition the plane into $r$ convex regions such that the following holds. For each region $C$ of the partition there is a subset of $c_r n^{1/r}$ lines of $A$ whose pairwise intersections are in $C$, and the same holds for $B$. In this statement $c_r$ only depends on $r$. We also prove that the dependence on $n$ is optimal.

math.MG↗

arXiv2019

Prescribing Symmetries and Automorphisms for Polytopes

We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When $Γ$ is a subgroup of the combinatorial automorphism group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with combinatorial automorphism group exactly $Γ$. When $Γ$ is a subgroup of the geometric symmetry group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with both geometric symmetry group and combinatorial automorphism group exactly $Γ$. These symmetry-breaking results then are applied to show that for every abelian group $Γ$ of even order and every involution $σ$ of $Γ$, there is a centrally symmetric convex polytope with geometric symmetry group $Γ$ such that $σ$ corresponds to the central symmetry.

math.CO↗

arXiv2019

Cutting a part from many measures

Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint. We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geq 2$, $d\geq 2$, $c\geq d$ with $m\geq n(c-d)+d$ are integers, and $μ_1, \dots, μ_m$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^d$, we prove that there exists a partition of $\mathbb{R}^d$ into $n$ convex pieces which equiparts the measures $μ_1, \dots, μ_{d-1}$, and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $μ_1, \dots, μ_m$.

math.CO↗

arXiv2018

Tverberg partitions as weak epsilon-nets

We prove a Tverberg-type theorem using the probabilistic method. Given $\varepsilon >0$, we find the smallest number of partitions of a set $X$ in $R^d$ into $r$ parts needed in order to induce at least one Tverberg partition on every subset of $X$ with at least $\varepsilon |X|$ elements. This generalizes known results about Tverberg's theorem with tolerance.

math.CO↗

arXiv2018

Tverberg's theorem is 50 years old: a survey

This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures.

math.CO↗

arXiv2017

Tverberg plus minus

We prove a Tverberg type theorem: Given a set $A \subset \mathbb{R}^d$ in general position with $|A|=(r-1)(d+1)+1$ and $k\in \{0,1,\ldots,r-1\}$, there is a partition of $A$ into $r$ sets $A_1,\ldots,A_r$ with the following property. The unique $z \in \bigcap_1^r \mathrm{aff} A_j$ can be written as an affine combination of the element in $A_j$: $z = \sum_{x \in A_j} α(x)x$ for every $p$ and exactly $k$ of the coefficients $α(x)$ are negative. The case $k=0$ is Tverberg's classical theorem.

math.GT↗

arXiv2017

Algorithms for Tverberg's theorem via centerpoint theorems

We obtain algorithms for computing Tverberg partitions based on centerpoint approximations. This applies to a wide range of convexity spaces, from the classic Euclidean setting to geodetic convexity in graphs. In the Euclidean setting, we present probabilistic algorithms which are weakly polynomial in the number of points and the dimension. For geodetic convexity in graphs, we obtain deterministic algorithms for cactus graphs and show that the general problem of finding the Radon number is NP-hard.

cs.CG↗

arXiv2017

Thieves can make sandwiches

We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of $m$ measures in $R^d$ among $r$ thieves using roughly $mr/d$ convex pieces, even in the cases when $m$ is larger than the dimension. The main proof relies on a construction of a geometric realization of the topological join of two spaces of partitions of $R^d$ into convex parts, and the computation of the Fadell-Husseini ideal valued index of the resulting spaces.

math.CO↗

arXiv2017

Robust Tverberg and colorful Carathéodory results via random choice

We use the probabilistic method to obtain versions of the colorful Carathéodory theorem and Tverberg's theorem with tolerance. In particular, we give bounds for the smallest integer $N=N(t,d,r)$ such that for any $N$ points in $R^d$, there is a partition of them into $r$ parts for which the following condition holds: after removing any $t$ points from the set, the convex hulls of what is left in each part intersect. We prove the bound $N=rt+O(\sqrt{t})$ for fixed $r,d$ which is polynomial in each parameters. Our bounds extend to colorful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.

math.MG↗

arXiv2016

Positive-fraction intersection results and variations of weak epsilon-nets

Given a finite set $X$ of points in $R^n$ and a family $F$ of sets generated by the pairs of points of $X$, we determine volumetric and structural conditions for the sets that allow us to guarantee the existence of a positive-fraction subfamily $F'$ of $F$ for which the sets have non-empty intersection. This allows us to show the existence of weak epsilon-nets for these families. We also prove a topological variation of the existence of weak epsilon-nets for convex sets.

math.MG↗

arXiv2016

Helly's Theorem: New Variations and Applications

This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization.

math.MG↗

arXiv2015

Quantitative $(p,q)$ theorems in combinatorial geometry

We show quantitative versions of classic results in discrete geometry, where the size of a convex set is determined by some non-negative function. We give versions of this kind for the selection theorem of Bárány, the existence of weak epsilon-nets for convex sets and the $(p,q)$ theorem of Alon and Kleitman. These methods can be applied to functions such as the volume, surface area or number of points of a discrete set. We also give general quantitative versions of the colorful Helly theorem for continuous functions.

math.MG↗

arXiv2015

Helly-type theorems for the diameter

We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional and $(p,q)$ versions work with conditions where the corresponding Helly theorem does not. We also include variants of Tverberg's theorem, Bárány's point selection theorem and the existence of weak epsilon-nets for convex sets with diameter estimates.

math.MG↗

arXiv2015

Computing the partition function for graph homomorphisms with multiplicities

We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain. As a corollary, we obtain quasi-polynomial algorithms for computing partition functions for independent sets, perfect matchings, Hamiltonian cycles and dense subgraphs in graphs as well as for graph colorings. This allows us to tell apart in quasi-polynomial time graphs that are sufficiently far from having a structure of a given type (i.e., independent set of a given size, Hamiltonian cycle, etc.) from graphs that have sufficiently many structures of that type, even when the probability to hit such a structure at random is exponentially small.

math.CO↗

arXiv2015

Computing the partition function for graph homomorphisms

We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries include efficient algorithms for computing weighted sums approximating the number of k-colorings and the number of independent sets in a graph, as well as an efficient procedure to distinguish pairs of edge-colored graphs with many color-preserving homomorphisms G --> H from pairs of graphs that need to be substantially modified to acquire a color-preserving homomorphism G --> H.

math.CO↗

arXiv2015

On a problem by Dol'nikov

In 2011 at an Oberwolfach workshop in Discrete Geometry, V. Dol'nikov posed the following problem. Consider three non-empty families of translates of a convex compact set $K$ in the plane. Suppose that every two translates from different families have a point of intersection. Is it always true that one of the families can be pierced by a set of three points? A result by R. N. Karasev from 2000 gives, in fact, an affirmative answer to the "monochromatic" version of the problem above. That is, if all the three families in the problem coincide. In the present paper we solve Dol'nikov's problem positively if $K$ is either centrally symmetric or a triangle, and show that the conclusion can be strengthened if $K$ is an euclidean disk. We also confirm the conjecture if we are given four families satisfying the conditions above.

math.MG↗