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Alfredo Hubard

Publications and source records attributed to Alfredo Hubard.

At least 19 recordsLinked to original sources
arXiv2026

Overlap-Helly theorems

In this paper we introduce a generalization of Helly's theorem closely connected to B\'ar\'any-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).

math.CO↗
arXiv2026

Volume and Projection Inequalities I: Zonoids and Courtade's Conjecture

We study volume and projection inequalities for zonoids through the multiaffine determinant polynomials that encode their volumes. We show that a log-submodularity conjecture for the volume of zonoids is equivalent to the Rayleigh property of zonotope volume polynomials, which we prove for the case when the degree or codegree is at most 3. This result is sharp in that when the degree and codegree are at least 4, we construct counterexamples using the existence of non-Rayleigh matroids in ranks at least four. Additionally, we provide unimodular or graphical counterexamples in dimensions four and higher to an equivalent projection inequality formulation of the conjecture. We also show that a stronger projection inequality fails already in dimension three. We next disprove Courtade's conjecture using a pair of orthogonal double bodies of revolution. Although Courtade's conjecture was originally formulated for general convex bodies, we show that it fails even for zonoids in every dimension at least three.

math.MG↗
arXiv2025

Crossing numbers of dense graphs on surfaces

In this paper, we provide upper and lower bounds on the crossing numbers of dense graphs on surfaces, which match up to constant factors. First, we prove that if $G$ is a dense enough graph with $m$ edges and $\Sigma$ is a surface of genus $g$, then any drawing of $G$ on $\Sigma$ incurs at least $\Omega \left(\frac{m^2}{g} \log ^2 g\right)$ crossings. The poly-logarithmic factor in this lower bound is new even in the case of complete graphs and disproves a conjecture of Shahrokhi, Sz\'ekely and Vrt'o from 1996. Then we prove a geometric converse to this lower bound: we provide an explicit family of hyperbolic surfaces such that for any graph $G$, sampling the vertices uniformly at random on this surface and connecting them with shortest paths yields $O\left(\frac{m^2}{g} \log ^2 g\right)$ crossings in expectation.

math.CO↗
arXiv2025

Crossing number inequalities for curves on surfaces

We prove that, as $m$ grows, any family of $m$ homotopically distinct closed curves on a surface induces a number of crossings that grows at least like $(m \log m)^2$. We use this to answer two questions of Pach, Tardos and Toth related to crossing numbers of drawings of multigraphs where edges are required to be non-homotopic. Furthermore, we generalize these results, obtaining effective bounds with optimal growth rates on every orientable surface.

math.GT↗
arXiv2025

Cubulating the sphere with many facets

For each $d\geq 3$ we construct cube complexes homeomorphic to the $d$-sphere with $n$ vertices in which the number of facets (assuming $d$ constant) is $\Omega(n^{5/4})$. This disproves a conjecture of Kalai's stating that the number of faces (of all dimensions) of cubical spheres is maximized by the boundaries of neighbourly cubical polytopes. The conjecture was already known to be false for $d=3$, $n=64$. Our construction disproves it for all $d\geq 3$ and $n$ sufficiently large. Moreover, since neighborly cubical polytopes have roughly $n (\log n)^{d/2}$ facets, we show that even the order of growth (at least for the number of facets) in the conjecture is wrong.

math.CO↗
arXiv2024

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 Sober\'on: \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↗
arXiv2024

Crossing lemmas for $k$-systems of arcs

We show a generalization of the crossing lemma for multi-graphs drawn on orientable surfaces in which pairs of edges are assumed to be drawn by non-homotopic simple arcs which pairwise cross at most $k$ times.

math.CO↗
arXiv2023

Weak $(1-ε)$-nets for polynomial superlevel sets

We prove that for any Borel probability measure $μ$ on $\mathbb R^n$ there exists a set $X\subset \mathbb R^n$ of $n+1$ points such that any $n$-variate quadratic polynomial $P$ that is nonnegative on $X$ (i.e. $P(x)\geq 0$, for every $x \in X$) satisfies $μ\{P\geq 0\}\geq \frac{2}{(n+1)(n+2)}$. We also prove that given an absolutely continuous probability measure $μ$ on $\mathbb R^n$ and $D\leq 2k$, for every $δ>0$ there exists a set $X\subset \mathbb R^n$ with $|X|\leq \binom{n+2k}{n}-n-1$ such that any $n$-variate polynomial $P$ of degree $D$ that is nonnegative on $X$ satisfies $μ\{P\geq 0\}\geq \frac{1}{\binom{n+2k}{n}+1} - δ$. These statements are analogues of the celebrated centerpoint theorem, which corresponds to the case of linear polynomials. Our results follow from new estimates on the Carathéodory numbers of real Veronese varieties, or alternatively, from bounds on the nonnegative symmetric rank of real symmetric tensors.

math.MG↗
arXiv2023

Degenerate crossing number and signed reversal distance

Given a graph drawn in the plane, the degenerate crossing number of the drawing is the number of points in the plane which are contained in the relative interior of at least two edges, where each edge is required to be drawn as a simple arc. The degenerate crossing number of a graph is the minimum degenerate crossing number among all its drawings. Given a drawing, cutting a neighborhood of the surface around each crossing and pasting a M\"obius band gives a non-orientable surface, on which the drawing of the graph can be extended to an embedding. From this observation, Mohar derived that the degenerate crossing number of a graph is at most its non-orientable genus, and conjectured that these quantities are equal for every graph. He also made a stronger conjecture for loopless pseudo-triangulations with a fixed embedding scheme. In this paper, we prove a structure theorem that allows to understand when the degenerate crossing number and non-orientable genus coincide in a large class of loopless bipartite embedding schemes. In particular, we provide a counterexample to Mohar's stronger conjecture, but show that in the vast majority of the 2-vertex cases, as well as for many bipartite graphs, Mohar's conjecture is satisfied. The reversal distance between two signed permutations is the minimum number of reversals that transform one permutation to the other one. If we represent the trajectory of each element of a signed permutation under successive reversals by a simple arc, we obtain a drawing of a 2-vertex embedding scheme with degenerate crossings. Our main result is proved by leveraging this connection and a classical result in genome rearrangement (the Hannenhalli--Pevzner algorithm) and can also be understood as an extension of this algorithm when the reversals do not necessarily happen in a monotone order.

cs.CG↗
arXiv2022

Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems

Given a complete simple topological graph $G$, a $k$-face generated by $G$ is the open bounded region enclosed by the edges of a non-self-intersecting $k$-cycle in $G$. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete $n$-vertex simple topological graph generates at least $Ω(n^{1/3})$ pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on $n$ vertices drawn in the unit square generates a 4-face with area at most $O(n^{-1/3})$. Finally, we investigate a $\mathbb Z_2$ variant of Heilbronn triangle problem.

math.CO↗
arXiv2022

Systolic inequalities for the number of vertices

Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of "essentiality", our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth--Nakamura cup-length systolic bound from manifolds to complexes.

math.MG↗
arXiv2022

Short Topological Decompositions of Non-Orientable Surfaces

In this article, we investigate short topological decompositions of non-orientable surfaces and provide algorithms to compute them. Our main result is a polynomial-time algorithm that for any graph embedded in a non-orientable surface computes a canonical non-orientable system of loops so that any loop from the canonical system intersects any edge of the graph in at most 30 points. The existence of such short canonical systems of loops was well known in the orientable case and an open problem in the non-orientable case. Our proof techniques combine recent work of Schaefer-Štefankovič with ideas coming from computational biology, specifically from the signed reversal distance algorithm of Hannenhalli-Pevzner. The existence of short canonical non-orientable systems of loops confirms a special case of a conjecture of Negami on the joint crossing number of two embeddable graphs. We also provide a correction for an argument of Negami bounding the joint crossing number of two non-orientable graph embeddings.

cs.CG↗
arXiv2021

Equipartitions and Mahler volumes of symmetric convex bodies

Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\mf for symmetric convex bodies}. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. We observe that these ideas give a large family of convex sets in every dimension for which the Mahler conjecture holds true. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a {\mf new} stability result.

math.MG↗
arXiv2019

On a topological version of Pach's overlap theorem

Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size $C(\log n)^{1/(d-1)}$. We show that this is tight in dimension $2$, for all surfaces other than $\mathbb{S}^2$. Surprisingly, the optimal bound for $\mathbb{S}^2$ in the topological version of Pach's theorem is of the order $(\log n)^{1/2}$. We conjecture that, among higher-dimensional manifolds, spheres are similarly distinguished. This improves upon the results of Bárány, Meshulam, Nevo and Tancer.

math.CO↗
arXiv2019

Random polytopes and the wet part for arbitrary probability distributions

We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in $\mathbf{R}^d$ relates to the wet part of that measure. This extends classical results for the uniform distribution from a convex set [Bárány and Larman 1988]. The lower bound of Bárány and Larman continues to hold in the general setting, but the upper bound must be relaxed by a factor of $\log n$. We show by an example that this is tight.

math.PR↗
arXiv2018

Limits of Order Types

We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erdős problem of finding the minimal density of 5-or 6-tuples in convex position in an arbitrary point set, and also an inequality expressing the difficulty of sampling order types uniformly. Next we establish results on the analytic representation of limits of order types by planar measures. Our main result is a rigidity theorem: we show that if sampling two measures induce the same probability distribution on order types, then these measures are projectively equivalent provided the support of at least one of them has non-empty interior. We also show that some condition on the Hausdorff dimension of the support is necessary to obtain projective rigidity and we construct limits of order types that cannot be represented by a planar measure. Returning to combinatorial geometry we relate the regularity of this analytic representation to the aforementioned problem of Erdős on the density of k-tuples in convex position, for large k.

math.CO↗
arXiv2018

Consistent sets of lines with no colorful incidence

We consider incidences among colored sets of lines in $\mathbb{R}^d$ and examine whether the existence of certain concurrences between lines of $k$ colors force the existence of at least one concurrence between lines of $k+1$ colors. This question is relevant for problems in 3D reconstruction in computer vision.

cs.CG↗