Discrete & Computational Geometry最新文献

We show that every complete n-vertex simple topological graph contains a topological subgraph on at least ((log n)^{1/4 - o(1)}) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound (Omega (log ^{1/8}n)) obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least ((log n)^{1 -o(1)}).

Given a set of points (P) and a set of regions (mathcal {O}), an incidence is a pair ((p,mathcalligra {o}) in Ptimes mathcal {O}) such that (pin mathcalligra {o}). We obtain a number of new results on a classical question in combinatorial geometry: What is the number of incidences (under certain restrictive conditions)? We prove a bound of (Obigl ( k n(log n/log log n)^{d-1} bigr )) on the number of incidences between n points and n axis-parallel boxes in (mathbb {R}^d), if no k boxes contain k common points, that is, if the incidence graph between the points and the boxes does not contain (K_{k,k}) as a subgraph. This new bound improves over previous work, by Basit et al. (Forum Math Sigma 9:59, 2021), by more than a factor of (log ^d n) for (d >2). Furthermore, it matches a lower bound implied by the work of Chazelle (J ACM 37(2):200–212, 1990), for (k=2), thus settling the question for points and boxes. We also study several other variants of the problem. For halfspaces, using shallow cuttings, we get a linear bound in two and three dimensions. We also present linear (or near linear) bounds for shapes with low union complexity, such as pseudodisks and fat triangles.

For a closed Riemannian manifold (mathcal {M}) and a metric space S with a small Gromov–Hausdorff distance to it, Latschev’s theorem guarantees the existence of a sufficiently small scale (beta >0) at which the Vietoris–Rips complex of S is homotopy equivalent to (mathcal {M}). Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale (beta ) in order to provide sampling conditions for S to be homotopy equivalent to (mathcal {M}). In this paper, we prove a stronger and pragmatic version of Latschev’s theorem, facilitating a simple description of (beta ) using the sectional curvatures and convexity radius of (mathcal {M}) as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris–Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris–Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds.

Equilateral triangles of sidelengths 1, (2^{-t}), (3^{-t}), (4^{-t},ldots ) can be packed perfectly into an equilateral triangle, provided that ( 1/2<t le 37/72). Moreover, for t slightly greater than 1/2, squares of sidelengths 1, (2^{-t}), (3^{-t}), (4^{-t},ldots ) can be packed perfectly into a square (S_t) in such a way that some squares have a side parallel to a diagonal of (S_t) and the remaining squares have a side parallel to a side of (S_t).

The Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set (widetilde{P}) of lifted points in the hyperbolic plane. Since (widetilde{P}) is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than (12g-6) with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.

We prove that the length of the projection of the vector joining the centers of mass of a convex body on the plane and of its boundary to an arbitrary direction does not exceed (frac{1}{6}) of the body width in this direction. It follows that the distance between these centers of mass does not exceed (frac{1}{6}) of the diameter of the body and (frac{1}{12}) of its boundary length. None of those constants can be improved.

We show that there exist convex bodies of constant width in ({mathbb {E}}^n) with illumination number at least ((cos (pi /14)+o(1))^{-n}), answering a question by Kalai. Furthermore, we prove the existence of finite sets of diameter 1 in ({mathbb {E}}^n) which cannot be covered by ((2/sqrt{3}-o(1))^{n}) balls of diameter 1, improving a result of Bourgain and Lindenstrauss.

We initiate the axiomatic study of affine oriented matroids (AOMs) on arbitrary ground sets, obtaining fundamental notions such as minors, reorientations and a natural embedding into the frame work of Complexes of Oriented Matroids. The restriction to the finitary case (FAOMs) allows us to study tope graphs and covector posets, as well as to view FAOMs as oriented finitary semimatroids. We show shellability of FAOMs and single out the FAOMs that are affinely homeomorphic to (mathbb {R}^n). Finally, we study group actions on AOMs, whose quotients in the case of FAOMs are a stepping stone towards a general theory of affine and toric pseudoarrangements. Our results include applications of the multiplicity Tutte polynomial of group actions of semimatroids, generalizing enumerative properties of toric arrangements to a combinatorially defined class of arrangements of submanifolds. This answers partially a question by Ehrenborg and Readdy.

We consider nine geometric systems: Miquel dynamics, P-nets, integrable cross-ratio maps, discrete holomorphic functions, orthogonal circle patterns, polygon recutting, circle intersection dynamics, (corrugated) pentagram maps and the short diagonal hyperplane map. Using a unified framework, for each system we prove an explicit expression for the solution as a function of the initial data; more precisely, we show that the solution is equal to the ratio of two partition functions of an oriented dimer model on an Aztec diamond whose face weights are constructed from the initial data. Then, we study the Devron property (Glick in J Geom Phys 87:161–189, 2015), which states the following: if the system starts from initial data that is singular for the backwards dynamics, this singularity is expected to reoccur after a finite number of steps of the forwards dynamics. Again, using a unified framework, we prove this Devron property for all of the above geometric systems, for different kinds of singular initial data. In doing so, we obtain new singularity results and also known ones (Glick in J Geom Phys 87:161–189, 2015; Yao in Glick’s conjecture on the point of collapse of axis-aligned polygons under the pentagram maps (2014). Preprint arXiv:1410.7806). Our general method consists in proving that these nine geometric systems are all related to the Schwarzian octahedron recurrence (dSKP equation), and then to rely on the companion paper (Affolter et al. in Comb Theory 3(2), 2023), where we study this recurrence in general, prove explicit expressions and singularity results.

We consider a geometric percolation process partially motivated by recent work of Hejda and Kala. Specifically, we start with an initial set (X subseteq {mathbb {Z}}^2), and then iteratively check whether there exists a triangle (T subseteq {mathbb {R}}^2) with its vertices in ({mathbb {Z}}^2) such that T contains exactly four points of ({mathbb {Z}}^2) and exactly three points of X. In this case, we add the missing lattice point of T to X, and we repeat until no such triangle exists. We study the limit sets S, the sets stable under this process, including determining their possible densities and some of their structure.