Author

Pablo Soberón

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Preprint Aug 2026

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 Gr\"unbaum's 1960 conjecture and disproves Ramos'general conjecture on hyperplane equipartitions. We reduce the construction to finding...

Pablo Soberón · 0 citations

Preprint Sep 2026

The topological B\'ar\'any-Larman conjecture for prime numbers

The B\'ar\'any-Larman conjecture states that for any $d+1$ sets of $r$ points each in $\mathbb{R}^d$, considered as color classes, we can partition their union into $r$ rainbow $(d+1)$-tuples whose convex hulls intersect. We prove that the topological version of this conjecture holds when $r$ is a prime number. We also...

Pablo Soberón · 0 citations

Preprint Aug 2026

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}...

Pablo Soberón · 0 citations