arXiv subjects

Pablo Soberón

Publications and source records attributed to Pablo Soberón.

59 records · Page 4Linked to original sources

arXiv2014

About an Erdős-Grünbaum conjecture concerning piercing of non bounded convex sets

In this paper, we study the number of compact sets needed in an infinite family of convex sets with a local intersection structure to imply a bound on its piercing number, answering a conjecture of Erdős and Grünbaum. Namely, if in an infinite family of convex sets in $\mathbb{R}^d$ we know that out of every $p$ there are $q$ which are intersecting, we determine if having some compact sets implies a bound on the number of points needed to intersect the whole family. We also study variations of this problem.

math.MG↗

arXiv2014

Measure Partitions Using Hyperplanes with Fixed Directions

We study nested partitions of $R^d$ obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing the parts among $k$ sets. This generalises classical necklace splitting results and their more recent high-dimensional versions. With similar methods we show that in the plane, for any $t$ measures there is a path formed only by horizontal and vertical segments using at most $t-1$ turns that splits them by half simultaneously, and optimal mass-partitioning results for chessboard-colourings of $R^d$ using hyperplanes with fixed directions.

math.MG↗

arXiv2012

Equal coefficients and tolerance in coloured Tverberg partitions

The coloured Tverberg theorem was conjectured by Bárány, Lovász and Füredi and asks whether for any d+1 sets (considered as colour classes) of k points each in R^d there is a partition of them into k colourful sets whose convex hulls intersect. This is known when d=1,2 or k+1 is prime. In this paper we show that (k-1)d+1 colour classes are necessary and sufficient if the coefficients in the convex combination in the colourful sets are required to be the same in each class. We also examine what happens if we want the convex hulls of the colourful sets to intersect even if we remove any r of the colour classes. Namely, if we have (r+1)(k-1)d+1 colour classes of k point each, there is a partition of them into k colourful sets such that they intersect using the same coefficients regardless of which r colour classes are removed. We also investigate the relation of the case k=2 and the Gale transform, obtaining a variation of the coloured Radon theorem.

math.MG↗

arXiv2012

An extension of a theorem by Yao & Yao

In this paper we study $N_d(k)$ the smallest positive integer such that any nice measure $μ$ in $\R^d$ can be partitioned in $N_d(k)$ parts of equal measure so that every hyperplane avoids at least $k$ of them. A theorem of Yao and Yao \cite{YY1985} states that $N_d(1) \le 2^d$. Among other results, we obtain the bounds $N_d(2) \le 3 \cdot 2^{d-1}$ and $N_d(1) \ge C \cdot 2^{d/2}$ for some constant $C$. We then apply these results to a problem on the separation of points and hyperplanes.

math.MG↗

arXiv2011

Balanced Convex Partitions of Measures in $\mathbb{R}^d$

We will prove the following generalization of the ham sandwich Theorem, conjectured by Imre Bárány. Given a positive integer $k$ and $d$ nice measures $μ_1, μ_2,..., μ_d$ in $\mathbb{R}^d$ such that $μ_i (\mathds{R}^d) = k$ for all $i$, there is a partition of $\mathbb{R}^d$ in $k$ interior-disjoint convex parts $C_1, C_2,..., C_k$ such that $μ_i (C_j) = 1$ for all $i,j$. If $k=2$ this gives the ham sandwich Theorem.

math.MG↗