19 papers
The topological Bárány-Larman conjecture for prime numbers
Pablo Soberón
The Bárány-Larman conjecture states that for any sets of points each in , considered as color classes, we can partition their union into rainbow …
Helly and Radon theorems for convex intersections containing -flats
Sarah Ludwigson, Pablo Soberón
We study versions of results in combinatorial geometry related to families of convex sets in whose intersection contains a -dimensional affine space. We prove gen…
Four hyperplanes do not always equipartition a mass in
Pablo Soberón
We construct a smooth strictly positive density in that cannot be divided into parts of the same size by four affine hyperplanes. This settles the last open cas…
Blocking codimension-one simplices on the moment curve
Pablo Soberón
We study , the minimum number of points needed to meet the relative interior of every -simplex spanned by an -point set in general position in . In…
Selection-structure generalizations of the Borsuk-Ulam theorem
Pablo Soberón
We prove Borsuk-Ulam-type results governed by selection structures. Selection structures extend the matroidal framework for colorful theorems in discrete geometry and include non-m…
KKM theorems and discrete geometry beyond matroids
Pablo Soberón
We introduce selection structures, a topological framework that extends the role played by color classes and matroids in discrete geometry and KKM theorems. Selection structures al…
Tverberg cores and Kalai's cascade conjecture
Pablo Soberón
We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an -simplex to , let be the set of points contained in the images…
Cookie cutters: Bisections with fixed shapes
Patrick Schnider, Pablo Soberón
In a mass partition problem, we are interested in finding equitable partitions of smooth measures in . In this manuscript, we study the problem of finding simultaneou…
Bisections of mass assignments by parallel hyperplanes
Nikola Sadovek, Pablo Soberón
In this paper, we prove a result on the bisection of mass assignments by parallel hyperplanes on Euclidean vector bundles. Our methods consist of the development of a novel lifting…
Complex analogues of the Tverberg--Vrećica conjecture and central transversal theorems
Nikola Sadovek, Pablo Soberón
The Tverberg--Vrećica conjecture claims a broad generalization of Tverberg's classical theorem. One of its consequences, the central transversal theorem, extends both the centerpoi…
Tverberg's theorem and multi-class support vector machines
Pablo Soberón
We show how, using linear-algebraic tools developed to prove Tverberg's theorem in combinatorial geometry, we can design new models of multi-class support vector machines (SVMs). T…
Extensions of discrete Helly theorems for boxes
Timothy Edwards, Pablo Soberón
We prove extensions of Halman's discrete Helly theorem for axis-parallel boxes in . Halman's theorem says that, given a set in , if is a finite…
Bisecting masses with families of parallel hyperplanes
Alfredo Hubard, Pablo Soberón
We prove a common generalization of several mass partition results using hyperplane arrangements to split into two sets. Our main result implies the ham sandwich the…
Improved Tverberg theorems for certain families of polytopes
Pablo Soberón, Shira Zerbib
A theorem of Grünbaum, which states that every -polytope is a refinement of an -simplex, implies the following generalization of Tverberg's theorem: if is a linear functi…
High-dimensional envy-free partitions
Pablo Soberón, Christina Yu
A vast array of envy-free results have been found for the subdivision of one-dimensional resources, such as the interval . The goal is to divide the space into pieces an…
Tverberg Partition Graphs
Deborah Oliveros, Érika Roldán, Pablo Soberón +1
Given a finite set of points in , Tverberg's theorem guarantees the existence of partitions of this set into parts whose convex hulls intersect. We introduce a graph…
Colorful and Quantitative Variations of Krasnosselsky's Theorem
Connor Donovan, Danielle Paulson, Pablo Soberón
Krasnosselsky's art gallery theorem gives a combinatorial characterization of star-shaped sets in Euclidean spaces, similar to Helly's characterization of finite families of convex…
Combinatorial Depth Measures for Hyperplane Arrangements
Patrick Schnider, Pablo Soberón
Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion that measures how good of a regression hyperplane a given query hyperplane is with respect to a set of dat…
Counterexamples to the Colorful Tverberg Conjecture for Hyperplanes
João Pedro Carvalho, Pablo Soberón
In 2008 Karasev conjectured that for every set of blue lines, green lines, and red lines in the plane, there exists a partition of them into colorful triples whose…