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Pablo Soberón

Publications and source records attributed to Pablo Soberón.

At least 19 recordsLinked to original sources
arXiv2026

Discrepancy theory, Tverberg's theorem, and regression depth

We prove new bounds for Tverberg's theorem with tolerance. We show that $N = rt+Θ_{d,r}(t^{1/2-1/(2d)})$, where $N$ is the smallest number such that any set of $N$ points in $\mathbb{R}^d$ has a partition into $r$ parts such that the convex hulls of the parts intersect even if we remove any $t$ of the points. We extend Tverberg's theorem with tolerance to families of hyperplanes in $\mathbb{R}^d$, and show that for any set of $rt + O_{d,r}(t^{1/2-1/(2d)}\sqrt{\log (t+1)})$ hyperplanes in $\mathbb{R}^d$ there exists a partition of them into $r$ parts such that the regression hulls of the parts intersect even if any $t$ hyperplanes are removed. Our bounds follow from establishing a connection between Tverberg-type results and discrepancy theory.

math.CO↗
arXiv2026

Four hyperplanes do not always equipartition a mass in $\mathbb{R}^4$

We construct a smooth strictly positive density in $\mathbb{R}^4$ that cannot be divided into $16$ parts of the same size by four affine hyperplanes. This settles the last open case of Grünbaum's 1960 conjecture and disproves Ramos' general conjecture on hyperplane equipartitions. We reduce the construction to finding two homogeneous polynomials in four variables, of degrees three and four, whose multilinear coefficients cannot vanish simultaneously after any orthogonal change of coordinates. We give two proofs of this nonvanishing result. The first uses a local perturbation argument. The second reduces it to the absence of a common zero for five explicit polynomials on $[-1,1]^6$, verified by a computer-assisted Bernstein subdivision argument.

math.CO↗
arXiv2026

KKM theorems and discrete geometry beyond matroids

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 allow us to extend classic results to genuinely non-matroidal examples, including chessboard complexes and matching complexes. We show that several matroidal versions of classic results can be generalized to selection structures. These include McGinnis' version of Komiya's KKMS theorem, Holmsen's version of Carathéodory's theorem, Kalai and Meshulam's version of Helly's theorem, and Sadovek's version of the Goodman--Pollack transversal theorem.

math.CO↗
arXiv2026

Helly and Radon theorems for convex intersections containing $k$-flats

We study versions of results in combinatorial geometry related to families of convex sets in $\mathbb{R}^d$ whose intersection contains a $k$-dimensional affine space. We prove generalizations of the colorful Radon theorem, the fractional Helly theorem, the colorful Helly theorem, and the selection-structure Helly theorem. When $k=0$, our arguments give new proofs of the corresponding versions for points.

math.CO↗
arXiv2026

Blocking codimension-one simplices on the moment curve

We study $b_d(n)$, the minimum number of points needed to meet the relative interior of every $(d-1)$-simplex spanned by an $n$-point set in general position in $\mathbb{R}^d$. In the plane, this is the parameter from the Blocking Conjecture. We improve the best known general planar lower bound to $ b_2(n)\ge \frac{41}{13}n-O\left(\frac{n}{\log n}\right)$. For $n$ points on the moment curve in even dimension $2r$, we prove that at least $\frac{1}{r!}n^r\log n-O_r(n^r)$ points are needed to pierce the relative interior of all its codimension-one simplices, which exceeds the number of codimension-one faces in a triangulation by a $\log n$ factor. For equally spaced points on the moment curve in odd dimensions, we construct an optimal blocking set whose size equals the maximum number of codimension-one faces in a triangulation.

math.CO↗
arXiv2026

Selection-structure generalizations of the Borsuk-Ulam theorem

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-matroidal examples such as chessboard complexes. Motivated by Frick and Wellner's Radon-type strengthening of Fan's theorem and its colorful variants, we prove selection-structure analogues whose conclusions are determined by Radon partitions. We also prove a prime-power selection-structure covering version of Volovikov's theorem, governed by Tverberg partitions. We include applications to fair partitions, including selection-structure versions of the ham sandwich and necklace splitting theorems.

math.CO↗
arXiv2026

Variations of Helly's theorem for convex splinters

A convex splinter $K$ is a union of convex sets in $\mathbb{R}^d$ such that every minimal affine dependent set of $\mathbb{R}^d$ contained in $K$ is contained in one of the sets. The study of convex splinters was motivated by the study of flat transversals to convex sets. We extend several variations of Helly's theorem from convex geometry to convex splinters. These include fractional and colorful variations of Helly's theorem. We also extend Tverberg's theorem to convex splinters.

math.CO↗
arXiv2026

Bisecting masses with families of parallel hyperplanes

We prove a common generalization of several mass partition results using hyperplane arrangements to split $\mathbb{R}^d$ into two sets. Our main result implies the ham sandwich theorem, the necklace splitting theorem for two thieves, a theorem about chessboard splittings using hyperplanes with fixed directions, and all known cases of Langerman's conjecture about bisections with $n$ hyperplanes. Our main result also confirms an infinite number of previously unknown cases of the following conjecture of Takahashi and Soberón: \emph{For any $d+k-1$ measures in $\mathbb{R}^d$, there exist an arrangement of $k$ parallel hyperplanes that bisects each of the measures.} The general result follows from the case of measures that are supported on a finite set with an odd number of points. The proof for this case is inspired by ideas of differential and algebraic topology, but it is a completely elementary parity argument. Additionally, we disprove a conjecture by Langerman on bisections of measures using hyperplane arrangements, showing that the conditions in our main result are sometimes necessary.

math.CO↗
arXiv2026

Tverberg cores and Kalai's cascade conjecture

We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an $n$-simplex to $\mathbb R^d$, let $T_r(f)$ be the set of points contained in the images of $r$ pairwise disjoint faces. We prove that if $r$ is a prime power and $\dim T_r(f)\le k$, then there exists a point that remains an $r$-Tverberg point after any $t$ vertices are deleted, provided $n=(r-1)(d+1)+t(k+1)$. For $t=1$, this gives a topological analogue of a standard consequence of Kalai's cascade conjecture. We also confirm the cascade conjecture for finite point sets whose Radon set is $0$-dimensional.

math.CO↗
arXiv2025

Complex analogues of the Tverberg--Vrećica conjecture and central transversal theorems

The Tverberg--Vrećica conjecture claims a broad generalization of Tverberg's classical theorem. One of its consequences, the central transversal theorem, extends both the centerpoint theorem and the ham sandwich theorem. In this manuscript, we establish complex analogues of these results, where the corresponding transversals are complex affine spaces. The proofs of the complex Tverberg--Vrećica conjecture and its optimal colorful version rely on the non-vanishing of an equivariant Euler class. Furthermore, we obtain new Borsuk--Ulam-type theorems on complex Stiefel manifolds. These theorems yield complex analogues of recent extensions of the ham sandwich theorem for mass assignments by Axelrod-Freed and Soberón, and provide a direct proof of the complex central transversal theorem.

math.CO↗
arXiv2025

Fair distribution of bundles

In this paper, we study the problem of splitting fairly bundles of items. We show that given $n$ bundles with $m$ kinds of items in them, it is possible to distribute the value of each kind of item fairly among $r$ persons by breaking apart at most $(r-1)m$ bundles. Moreover, we can guarantee that each participant will receive roughly $n/r - mr/2$ full bundles. The proof methods are topological and use a modified form of the configuration space/test map scheme. We obtain optimal results when $r$ is a power of two.

math.CO↗
arXiv2025

Partitions of mass assignments with spheres and wedges

In this paper, we generalize classic mass partition results dealing with partitions using spheres, parallel hyperplanes, or axis-parallel wedges to the setting of mass assignments. In a mass assignment problem, we assign mass distributions continuously to all $k$-dimensional subspaces of $\mathbb{R}^d$, and seek to guarantee the existence of a particular subspace in which more masses can be bisected than those by analyzing the problem in $\mathbb{R}^k$. We prove new mass assignment results for spheres, parallel hyperplanes, and axis-parallel wedges. The proof techniques rely on new Borsuk--Ulam type theorems on spheres and Stiefel manifolds.

math.CO↗
arXiv2025

Bisections of mass assignments by parallel hyperplanes

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 method to define the configuration space--test map scheme, which transforms the problem to a Borsuk--Ulam-type question on equivariant fiber bundles, along with a new computation of the parametrized Fadell--Husseini index. As the primary application, we show that any $d+k+m-1$ mass assignments to linear $d$-spaces in $\mathbb{R}^{d+m}$ can be bisected by $k$ parallel hyperplanes in at least one $d$-space, provided that the Stirling number of the second kind $S(d+k+m-1, k)$ is odd. This generalizes all known cases of a conjecture by Soberón and Takahashi, which asserts that any $d+k-1$ measures in $\mathbb{R}^d$ can be bisected by $k$ parallel hyperplanes.

math.AT↗
arXiv2025

Bisections of mass assignments by parallel hyperplanes

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 method to define the configuration space--test map scheme, which transforms the problem to a Borsuk--Ulam-type question on equivariant fiber bundles, along with a new computation of the parametrized Fadell--Husseini index. As the primary application, we show that any $d+k+m-1$ mass assignments to linear $d$-spaces in $\mathbb{R}^{d+m}$ can be bisected by $k $ parallel hyperplanes in at least one $d$-space, provided that the Stirling number of the second kind $S(d+k+m-1, k)$ is odd. This generalizes all known cases of a conjecture by Soberón and Takahashi, which asserts that any $d+k-1$ measures in $\mathbb{R}^d$ can be bisected by $k$ parallel hyperplanes.

math.AT↗
arXiv2025

Cookie cutters: Bisections with fixed shapes

In a mass partition problem, we are interested in finding equitable partitions of smooth measures in $\mathbb{R}^d$. In this manuscript, we study the problem of finding simultaneous bisections of measures using scaled copies of a prescribed set $K$. We distinguish the problem when we are allowed to use scaled and translated copies of $K$ and the problem when we are allowed to use scaled isometric copies of $K$. These problems have only previously been studied if $K$ is a half-space or a Euclidean ball. We obtain positive results for simultaneous bisection of any $d+1$ masses for star-shaped compact sets $K$ with non-empty interior, where the conditions on the problem depend on the smoothness of the boundary of $K$. Additional proofs are included for particular instances of $K$, such as hypercubes and cylinders, answering positively a conjecture of Soberón and Takahashi. The proof methods are topological and involve new Borsuk--Ulam-type theorems.

math.CO↗
arXiv2024

Improved Tverberg theorems for certain families of polytopes

A theorem of Grünbaum, which states that every $m$-polytope is a refinement of an $m$-simplex, implies the following generalization of Tverberg's theorem: if $f$ is a linear function from an $m$-dimensional polytope $P$ to $\mathbb{R}^d$ and $m \ge (d + 1)(r - 1)$, then there are $r$ pairwise disjoint faces of $P$ whose images intersect. Moreover, the topological Tverberg theorem implies that this statement is true whenever the map $f$ is continuous and $r$ is a prime power. In this note, we show that for certain families of polytopes the lower bound on the dimension $m$ of the polytopes can be significantly improved, both in the affine and topological cases.

math.CO↗
arXiv2024

Tverberg's theorem and multi-class support vector machines

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). These supervised learning protocols require fewer conditions to classify sets of points, and can be computed using existing binary SVM algorithms in higher-dimensional spaces, including soft-margin SVM algorithms. We describe how the theoretical guarantees of standard support vector machines transfer to these new classes of multi-class support vector machines. We give a new simple proof of a geometric characterization of support vectors for largest margin SVMs by Veelaert.

cs.LG↗