arXiv subjects

Jiří Matoušek

Publications and source records attributed to Jiří Matoušek.

24 records · Page 2Linked to original sources

arXiv2009

Zone Diagrams in Euclidean Spaces and in Other Normed Spaces

Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.

cs.CG↗

arXiv2009

Vectors in a Box

For an integer d>=1, let tau(d) be the smallest integer with the following property: If v1,v2,...,vt is a sequence of t>=2 vectors in [-1,1]^d with v1+v2+...+vt in [-1,1]^d, then there is a subset S of {1,2,...,t} of indices, 2<=|S|<=tau(d), such that \sum_{i\in S} vi is in [-1,1]^d. The quantity tau(d) was introduced by Dash, Fukasawa, and Günlük, who showed that tau(2)=2, tau(3)=4, and tau(d)=Omega(2^d), and asked whether tau(d) is finite for all d. Using the Steinitz lemma, in a quantitative version due to Grinberg and Sevastyanov, we prove an upper bound of tau(d) <= d^{d+o(d)}, and based on a construction of Alon and Vu, whose main idea goes back to Hastad, we obtain a lower bound of tau(d)>= d^{d/2-o(d)}. These results contribute to understanding the master equality polyhedron with multiple rows defined by Dash et al., which is a "universal" polyhedron encoding valid cutting planes for integer programs (this line of research was started by Gomory in the late 1960s). In particular, the upper bound on tau(d) implies a pseudo-polynomial running time for an algorithm of Dash et al. for integer programming with a fixed number of constraints. The algorithm consists in solving a linear program, and it provides an alternative to a 1981 dynamic programming algorithm of Papadimitriou.

math.CO↗

arXiv2009

Lower bounds for weak epsilon-nets and stair-convexity

A set N is called a "weak epsilon-net" (with respect to convex sets) for a finite set X in R^d if N intersects every convex set that contains at least epsilon*|X| points of X. For every fixed d>=2 and every r>=1 we construct sets X in R^d for which every weak (1/r)-net has at least Omega(r log^{d-1} r) points; this is the first superlinear lower bound for weak epsilon-nets in a fixed dimension. The construction is a "stretched grid", i.e., the Cartesian product of d suitable fast-growing finite sequences, and convexity in this grid can be analyzed using "stair-convexity", a new variant of the usual notion of convexity. We also consider weak epsilon-nets for the diagonal of our stretched grid in R^d, d>=3, which is an "intrinsically 1-dimensional" point set. In this case we exhibit slightly superlinear lower bounds (involving the inverse Ackermann function), showing that upper bounds by Alon, Kaplan, Nivasch, Sharir, and Smorodinsky (2008) are not far from the truth in the worst case. Using the stretched grid we also improve the known upper bound for the so-called "second selection lemma" in the plane by a logarithmic factor: We obtain a set T of t triangles with vertices in an n-point set in the plane such that no point is contained in more than O(t^2 / (n^3 log (n^3/t))) triangles of T.

math.CO↗

arXiv2009

Hardness of embedding simplicial complexes in $\R^d$

Let EMBED(k,d) be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R^d? Known results easily imply polynomiality of EMBED(k,2) (k=1,2; the case k=1, d=2 is graph planarity) and of EMBED(k,2k) for all k>2 (even if k is not considered fixed). We observe that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBED(d,d) and EMBED(d-1,d) are undecidable for each d>4. Our main result is NP-hardness of EMBED(2,4) and, more generally, of EMBED(k,d) for all k,d with d>3 and d\geq k \geq (2d-2)/3. These dimensions fall outside the so-called metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spież, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.

cs.CG↗

arXiv2008

Stabbing simplices by points and flats

The following result was proved by Barany in 1982: For every d >= 1 there exists c_d > 0 such that for every n-point set S in R^d there is a point p in R^d contained in at least c_d n^{d+1} - O(n^d) of the simplices spanned by S. We investigate the largest possible value of c_d. It was known that c_d <= 1/(2^d(d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c_d <= (d+1)^{-(d+1)}, and we conjecture this estimate to be tight. The best known lower bound, due to Wagner, is c_d >= gamma_d := (d^2+1)/((d+1)!(d+1)^{d+1}); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than gamma_d n^{d+1}+O(n^d) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved. We also prove that for every n-point set S in R^d there exists a (d-2)-flat that stabs at least c_{d,d-2} n^3 - O(n^2) of the triangles spanned by S, with c_{d,d-2}>=(1/24)(1- 1/(2d-1)^2). To this end, we establish an equipartition result of independent interest (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in R^d can be divided into 4d-2 equal parts by 2d-1 hyperplanes intersecting in a common (d-2)-flat.

math.CO↗

arXiv1998

Almost-tiling the plane by ellipses

For any delta > 1 we construct a periodic and locally finite packing of the plane with ellipses whose delta-enlargement covers the whole plane. This answers a question of Imre Bárány. On the other hand, we show that if C is a packing in the plane with circular discs of radius at most 1, then its 1.00001-enlargement covers no square with side length 4.

math.MG↗