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Luis Montejano

Publications and source records attributed to Luis Montejano.

36 records · Page 2Linked to original sources
arXiv2019

Different versions of the nerve theorem and rainbow simplices

Given a simplicial complex and a collection of subcomplexes covering it, the nerve theorem, a fundamental tool in topological combinatorics, guarantees a certain connectivity of the simplicial complex when connectivity conditions on the intersection of the subcomplexes are satisfied. We show that it is possible to extend this theorem by replacing some of these connectivity conditions on the intersection of the subcomplexes by connectivity conditions on their union. While this is interesting for its own sake, we use this extension to generalize in various ways the Meshulam lemma, a powerful homological version of the Sperner lemma. We also prove a generalization of the Meshulam lemma that is somehow reminiscent of the polytopal generalization of the Sperner lemma by De Loera, Peterson, and Su. For this latter result, we use a different approach and we do not know whether there is a way to get it via a nerve theorem of some kind.

math.CO↗
arXiv2019

The graphs behind Reuleaux polyhedra

This work is about graphs arising from Reuleaux polyhedra. Such graphs must necessarily be planar, $3$-connected and strongly self-dual. We study the question of when these conditions are sufficient. If $G$ is any such a graph with isomorphism $τ: G \to G^*$ (where $G^*$ is the unique dual graph), a metric mapping is a map $η: V(G) \to \mathbb R^3$ such that the diameter of $η(G)$ is $1$ and for every pair of vertices $(u,v)$ such that $u\in τ(v)$ we have dist$(η(u),η(v)) = 1$. If $η$ is injective, it is called a metric embedding. Note that a metric embedding gives rise to a Reuleaux Polyhedra. Our contributions are twofold: Firstly, we prove that any planar, $3$-connected, strongly self-dual graph has a metric mapping by proving that the chromatic number of the diameter graph (whose vertices are $V(G)$ and whose edges are pairs $(u,v)$ such that $u\in τ(v)$) is at most $4$, which means there exists a metric mapping to the tetrahedron. Furthermore, we use the Lovász neighborhood-complex theorem in algebraic topology to prove that the chromatic number of the diameter graph is exactly $4$. Secondly, we develop algorithms that allow us to obtain every such graph with up to $14$ vertices. Furthermore, we numerically construct metric embeddings for every such graph. From the theorem and this computational evidence we conjecture that every such graph is realizable as a Reuleaux polyhedron in $\mathbb R^3$. In previous work the first and last authors described a method to construct a constant-width body from a Reuleaux polyhedron. So in essence, we also construct hundreds of new examples of constant-width bodies. This is related to a problem of Vázsonyi, and also to a problem of Blaschke-Lebesgue.

cs.CG↗
arXiv2017

Codimension two and three Kneser Transversals

Let $k,d,λ\geqslant 1$ be integers with $d\geqslant λ$ and let $X$ be a finite set of points in $\mathbb{R}^{d}$. A $(d-λ)$-plane $L$ transversal to the convex hulls of all $k$-sets of $X$ is called Kneser transversal. If in addition $L$ contains $(d-λ)+1$ points of $X$, then $L$ is called complete Kneser transversal.In this paper, we present various results on the existence of (complete) Kneser transversals for $λ=2,3$. In order to do this, we introduce the notions of stability and instability for (complete) Kneser transversals. We first give a stability result for collections of $d+2(k-λ)$ points in $\mathbb{R}^d$ with $k-λ\geqslant 2$ and $λ=2,3$. We then present a description of Kneser transversals $L$ of collections of $d+2(k-λ)$ points in $\mathbb{R}^d$ with $k-λ\geqslant 2$ for $λ=2,3$. We show that either $L$ is a complete Kneser transversal or it contains $d-2(λ-1)$ points and the remaining $2(k-1)$ points of $X$ are matched in $k-1$ pairs in such a way that $L$ intersects the corresponding closed segments determined by them. The latter leads to new upper and lower bounds (in the case when $λ=2$ and $3$) for $m(k,d,λ)$ defined as the maximum positive integer $n$ such that every set of $n$ points (not necessarily in general position) in $\mathbb{R}^{d}$ admit a Kneser transversal.Finally, by using oriented matroid machinery, we present some computational results (closely related to the stability and unstability notions). We determine the existence of (complete) Kneser transversals for each of the $246$ different order types of configurations of $7$ points in $\mathbb{R}^3$.

math.CO↗
arXiv2017

Shadows of a Closed Curve

A shadow of a geometric object $A$ in a given direction $v$ is the orthogonal projection of $A$ on the hyperplane orthogonal to $v$. We show that any topological embedding of a circle into Euclidean $d$-space can have at most two shadows that are simple paths in linearly independent directions. The proof is topological and uses an analog of basic properties of degree of maps on a circle to relations on a circle. This extends a previous result which dealt with the case $d=3$.

math.MG↗
arXiv2016

Meissner Polyhedra

In this paper we develop a concrete way to construct bodies of constant width in dimension three. They are constructed from special embeddings of self-dual graphs.

math.MG↗
arXiv2016

Rotors in triangles and tethrahedra

A polytope $P$ is circumscribed about a convex body $Φ\subset \mathbb{R}^n$ if $Φ\subset P$ and each facet of $P$ is contained in a support hyperplane of $Φ$. We say that a convex body $Φ\subset \mathbb{R}^n$ is a rotor of a polytope $P$ if for each rotation $ρ$ of $\mathbb{R}^n$ there exist a translation $τ$ so that $P$ is circumscribed about $τρΦ$. In this paper we shall prove that if $P$ is a triangle, then there is a baricentric formula that describes the curvature of bd$Φ$ at the contact points, $\{A_1, A_2,A_3\}$. We prove also that if $Φ\subset \mathbb{R}^3$ is a convex body which is a rotor in a tetrahedron $T$ and if $Φ$ intersects the faces of $T$ at the points $\{x_1, \dots, x_4\}$, then the normal lines of $Φ$ at the contact points with $T$, $\{x_1, \dots, x_4\}$ generically belong to one ruling of a quadric surface.

math.MG↗
arXiv2016

Complete Kneser Transversals

Let $k,d,λ\geqslant1$ be integers with $d\geqslantλ$. Let $m(k,d,λ)$ be the maximum positive integer $n$ such that every set of $n$ points (not necessarily in general position) in $\mathbb{R}^{d}$ has the property that the convex hulls of all $k$-sets have a common transversal $(d-λ)$-plane. It turns out that $m(k, d,λ)$ is strongly connected with other interesting problems, for instance, the chromatic number of Kneser hypergraphs and a discrete version of Rado's centerpoint theorem. In the same spirit, we introduce a natural discrete version $m^*$ of $m$ by considering the existence of complete Kneser transversals. We study the relation between them and give a number of lower and upper bounds of $m^*$ as well as the exact value in some cases. The main ingredient for the proofs are Radon's partition theorem as well as oriented matroids tools. By studying the alternating oriented matroid we obtain the asymptotic behavior of the function $m^*$ for the family of cyclic polytopes.

math.CO↗
arXiv2015

On transversal and $2$-packing numbers in straight line systems on $\mathbb{R}^{2}$

A linear system is a pair $(X,\mathcal{F})$ where $\mathcal{F}$ is a finite family of subsets on a ground set $X$, and it satisfies that $|A\cap B|\leq 1$ for every pair of distinct subsets $A,B \in \mathcal{F}$. As an example of a linear system are the straight line systems, which family of subsets are straight line segments on $\mathbb{R}^{2}$. By $τ$ and $ν_2$ we denote the size of the minimal transversal and the 2--packing numbers of a linear system respectively. A natural problem is asking about the relationship of these two parameters; it is not difficult to prove that there exists a quadratic function $f$ holding $τ\leq f(ν_2)$. However, for straight line system we believe that $τ\leqν_2-1$. In this paper we prove that for any linear system with $2$-packing numbers $ν_2$ equal to $2, 3$ and $4$, we have that $τ\leqν_2$. Furthermore, we prove that the linear systems that attains the equality have transversal and $2$-packing numbers equal to $4$, and they are a special family of linear subsystems of the projective plane of order $3$. Using this result we confirm that all straight line systems with $ν_2\in\{2,3,4\}$ satisfies $τ\leqν_2-1$.

math.CO↗
arXiv2015

A geometric Hall-type theorem

We introduce a geometric generalization of Hall's marriage theorem. For any family $F = \{X_1, \dots, X_m\}$ of finite sets in $\mathbb{R}^d$, we give conditions under which it is possible to choose a point $x_i\in X_i$ for every $1\leq i \leq m$ in such a way that the points $\{x_1,...,x_m\}\subset \mathbb{R}^d$ are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell's celebrated generalization of Hall's theorem.

math.CO↗
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

Convex bodies with many elliptic sections

{We show in this paper that two normal elliptic sections through every point of the boundary of a smooth convex body essentially characterize an ellipsoid and furthermore, that four different pairwise non-tangent elliptic sections through every point of the $C^2$-differentiable boundary of a convex body also essentially characterize an ellipsoid.

math.MG↗
arXiv2014

The (p,q)-extremal problem and the fractional chromatic number of Kneser hypergraphs

The problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The \emph{$(p,q)$-extremal problem} consists in finding the maximum number of edges on a $k$-uniform hypergraph $\mathcal{H}$ with $n$ vertices such that among any $p$ edges some $q$ of them have no empty intersection. In this paper we have found a link between the fractional chromatic number of Kneser hypergraphs and the $(p,q)$-extremal problem and also solve the $(p,q)$-extremal problem for graphs if $n$ is sufficiently large and $p \geq q \geq 3$ by proposing it as a problem of extremal graph theory. With the aid of this result we calculate the fractional chromatic number of Kneser hypergraphs when they are composed with sets of cardinality 2.

math.CO↗
arXiv2014

A New Topological Helly Theorem and some Transversals Results

We prove that for a topological space X with the property that $H_p(U)=0$ for $p\geq d$ and every open subset $U$ of $X$, a finite family of open sets in $X$ has nonempty intersection if for any subfamily of size $j$, $1\leq j \leq d+1$, the $(d-j)$-dimensional homology group of its intersection is zero. We use this theorem to prove new results concerning transversal affine planes to families of convex sets.

math.MG↗
arXiv2014

Straight Line motion with rigid sets

If one is given a rigid triangle in the plane or space, we show that the only motion possible, where each vertex of the triangle moves along a straight line, is given by a hypocycloid line drawer in the plane, and a natural extension in three-space. Each point lies on a circle which rolls around, without slipping, inside a larger circle of twice its diameter.

math.MG↗
arXiv2012

Transitive oriented 3-Hypergraphs of cyclic orders

In this paper we introduce the definition of transitivity for oriented 3-hypergraphs in order to study partial and complete cyclic orders. This definition allow us to give sufficient conditions on a partial cyclic order to be totally extendable. Furthermore, we introduce the 3-hypergraph associated to a cyclic permutation and characterize it in terms of cyclic comparability 3-hypergraphs.

math.CO↗
arXiv2010

Order Types of convex bodies

We give new bounds on the Erdos-Szekeres theorems for convex bodies of Bisztriczky and Fejes Toth and of Pach and Toth. We derive them from a combinatorial characterization of convex position of a family of planar convex bodies. This characterization confirms that the concept of Order Type for points can be extended to noncrossing families of convex bodies in a geometrically meaningful way.

math.CO↗