Discrete & Computational Geometry最新文献

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers (kge 0) and (nge 1) we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as ({textbf{D}}_{n,k}^{textrm{VR}}). We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which ({textbf{D}}_{4,1}^{textrm{VR}}) fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality (2k+2) with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction.

Let ({{mathscr {A}}}) be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction ({{mathscr {A}}}'') of ({{mathscr {A}}}) to any hyperplane endowed with the natural multiplicity (kappa ) is then a free multiarrangement (({{mathscr {A}}}'',kappa )). The aim of this paper is to prove an analogue of Ziegler’s theorem for the stronger notion of inductive freeness: if ({{mathscr {A}}}) is inductively free, then so is the multiarrangement (({{mathscr {A}}}'',kappa )). In a related result we derive that if a deletion ({{mathscr {A}}}') of ({{mathscr {A}}}) is free and the corresponding restriction ({{mathscr {A}}}'') is inductively free, then so is (({{mathscr {A}}}'',kappa ))—irrespective of the freeness of ({{mathscr {A}}}). In addition, we show counterparts of the latter kind for additive and recursive freeness.

We examine two natural operations to create numerical semigroups. We say that a numerical semigroup ({mathcal {S}}) is k-normalescent if it is the projection of the set of integer points in a k-dimensional polyhedral cone, and we say that ({mathcal {S}}) is a k-quotient if it is the quotient of a numerical semigroup with k generators. We prove that all k-quotients are k-normalescent, and although the converse is false in general, we prove that the projection of the set of integer points in a cone with k extreme rays (possibly lying in a dimension smaller than k) is a k-quotient. The discrete geometric perspective of studying cones is useful for studying k-quotients: in particular, we use it to prove that the sum of a (k_1)-quotient and a (k_2)-quotient is a ((k_1+k_2))-quotient. In addition, we prove several results about when a numerical semigroup is not k-normalescent.

We establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in (mathbb {R}^3) and (mathbb {R}^4). In dimension three this distance is equal to (frac{9}{5}), and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.

Kalai conjectured that if P is a simplicial d-polytope that has no missing faces of dimension (d-1), then the graph of P and the space of affine 2-stresses of P determine P up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if (2le ile d/2) and P is a simplicial d-polytope that has no missing faces of dimension (ge d-i+1), then the space of affine i-stresses of P determines the space of affine 1-stresses of P. We prove this conjecture for (1) k-stacked d-polytopes with (2le ile kle d/2-1), (2) d-polytopes that have no missing faces of dimension (ge d-2i+2), and (3) flag PL ((d-1))-spheres with generic embeddings (for all (2le ile d/2)). We also discuss several related results and conjectures. For instance, we show that if P is a simplicial d-polytope that has no missing faces of dimension (ge d-2i+2), then the ((i-1))-skeleton of P and the set of sign vectors of affine i-stresses of P determine the combinatorial type of P. Along the way, we establish the partition of unity of affine stresses: for any (1le ile (d-1)/2), the space of affine i-stresses of a simplicial d-polytope as well as the space of affine i-stresses of a simplicial ((d-1))-sphere (with a generic embedding) can be expressed as the sum of affine i-stress spaces of vertex stars. This is analogous to Adiprasito’s partition of unity of linear stresses for Cohen–Macaulay complexes.

Let (P) be a set of n points in (mathbb {R}^2). For a parameter (varepsilon in (0,1)), a subset (Csubseteq P) is an (varepsilon )-kernel of (P) if the projection of the convex hull of (C) approximates that of (P) within ((1-varepsilon ))-factor in every direction. The set (C) is a weak (varepsilon )-kernel of (P) if its directional width approximates that of (P) in every direction. Let (textsf{k}_{varepsilon }(P)) (resp. (textsf{k}^{textsf{w}}_{varepsilon }(P))) denote the minimum-size of an (varepsilon )-kernel (resp. weak (varepsilon )-kernel) of (P). We present an (O(ntextsf{k}_{varepsilon }(P)log n))-time algorithm for computing an (varepsilon )-kernel of (P) of size (textsf{k}_{varepsilon }(P)), and an (O(n^2log n))-time algorithm for computing a weak (varepsilon )-kernel of (P) of size (textsf{k}^{textsf{w}}_{varepsilon }(P)). We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of (varepsilon )-core, a convex polygon lying inside , prove that it is a good approximation of the optimal (varepsilon )-kernel, present an efficient algorithm for computing it, and use it to compute an (varepsilon )-kernel of small size.

Let K be a convex body (a compact convex set) in (mathbb {R}^d), that contains a copy of another body S in every possible orientation. Is it always possible to continuously move any one copy of S into another, inside K? As a stronger question, is it always possible to continuously select, for each orientation, one copy of S in that orientation? These questions were asked by Croft. We show that, in two dimensions, the stronger question always has an affirmative answer. We also show that in three dimensions the answer is negative, even for the case when S is a line segment – but that in any dimension the first question has a positive answer when S is a line segment. And we prove that, surprisingly, the answer to the first question is negative in dimensions four and higher for general S.

A family of fractal arrangements of circles is introduced for each imaginary quadratic field K. Collectively, these arrangements contain (up to an affine transformation) every set of circles in the extended complex plane with integral curvatures and Zariski dense symmetry group. When that set is a circle packing, we show how the ambient structure of our arrangement gives a geometric criterion for satisfying the asymptotic local–global principle. Connections to the class group of K are also explored. Among them is a geometric property that guarantees certain ideal classes are group generators.

Finding point configurations, that yield the maximum polarization (Chebyshev constant) is gaining interest in the field of geometric optimization. In the present article, we study the problem of unconstrained maximum polarization on compact sets. In particular, we discuss necessary conditions for local optimality, such as that a locally optimal configuration is always contained in the convex hull of the respective darkest points. Building on this, we propose two sequences of mixed-integer linear programs in order to compute lower and upper bounds on the maximal polarization, where the lower bound is constructive. Moreover, we prove the convergence of these sequences towards the maximal polarization.

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a “giant component” of vertices, with measure (1-o(1)) and polynomial diameter. Both bounds rely on spectral gaps—of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second—which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.