An \((n+1)\)-toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group. Toroids are generalisations of maps on the torus to higher dimensions and also provide examples of abstract polytopes. Equivelar toroids are those that are induced by regular tessellations. In this paper we present a classification of equivelar \((n+1)\)-toroids with at most n flag-orbits;

For fixed k we prove exponential lower bounds on the equilateral number of subspaces of \(\ell _{\infty }^n\) of codimension k. In particular, we show that subspaces of codimension 2 of \(\ell _{\infty }^{n+2}\) and subspaces of codimension 3 of \(\ell _{\infty }^{n+3}\) have an equilateral set of cardinality \(n+1\) if \(n\ge 7\) and \(n\ge 12\) respectively. Moreover, the same is true for every normed

The lattice size of a lattice polytope P was defined and studied by Schicho, and Castryck and Cools. They provided an “onion skins” algorithm for computing the lattice size of a lattice polygon P in \(\mathbb R^2\) based on passing successively to the convex hull of the interior lattice points of P. We explain the connection of the lattice size to the successive minima of \(K=(P+(-P))^*\) and to the

We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold M. Along the way, we show that the Reeb number \(\mathcal {R}(M)\), i.e., the maximum cycle rank among all Reeb graphs of functions on M, is equal to the corank of fundamental group \(\pi _1(M)\), thus

It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e., the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representations of certain gain graphs.

This paper deals with visibility problems in Euclidean spaces where the set of obstacles Y is an infinite discrete point set. We prove five independent results. Consider the following problem. Given \(\varepsilon >0\), imagine a forest whose trees have radius \(\varepsilon \) and their locations are given by the set Y. Suppose that a light source is at infinity, and that there are no arbitrarily large

We establish the existence of arbitrary-length arithmetic progressions in model sets and Meyer sets in Euclidean d-space. We prove a van der Waerden-type theorem for Meyer sets. We show that subsets of Meyer sets with positive density and pure point diffraction contain arithmetic progressions of arbitrary length.

We show that the lattice Hadwiger (= kissing) number of superballs is exponential in the dimension. The same methods can be used to show exponential growth for more general convex bodies as well.

We investigate the question of whether any d-colorable simplicial d-polytope can be octahedralized, i.e., can be subdivided to a d-dimensional geometric cross-polytopal complex. We give a positive answer in dimension 3, with the additional property that the octahedralization introduces no new vertices on the boundary of the polytope.

The geometric problem of estimating an unknown compact convex set from evaluations of its support function arises in a range of scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the error over all possible compact convex sets; in particular, these methods allow for limited incorporation of prior structural information about the underlying set

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. A graph G is k-crossing-critical if its crossing number is at least k, but if we remove any edge of G, its crossing number drops below k. There are examples of k-crossing-critical graphs that do not have drawings with exactly k crossings. Richter and Thomassen proved in 1993 that if G is k-crossing-critical

We quantise Whitney’s construction to prove the existence of a triangulation for any \(C^2\) manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.

We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The estimate is achieved by means of obtaining new linear constraints on the autocorrelation function of A utilizing triple-order correlations in A, a concept that has not

Let G be a directed graph with n vertices and m edges, embedded on a surface S, possibly with boundary, with first Betti number \(\beta \). We consider the complexity of finding closed directed walks in G that are either contractible (trivial in homotopy) or bounding (trivial in integer homology) in S. Specifically, we describe algorithms to determine whether G contains a simple contractible cycle

We show that if a finite non-collinear set of points in \(\mathbb {C}^2\) lies on a family of m concurrent lines, and if one of those lines contains more than \(m-2\) points, there exists a line passing through exactly two points of the set. The bound \(m-2\) in our result is optimal. Our main theorem resolves a conjecture of Frank de Zeeuw, and generalizes a result of Kelly and Nwankpa.

We prove an extension of a ham sandwich theorem for families of lines in the plane by Dujmović and Langerman. Given two sets A, B of n lines each in the plane, we prove that it is possible to partition the plane into r closed convex regions so that the following holds. For each region C of the partition there is a subset of \(c_r n^{1/r}\) lines of A whose pairwise intersections are in C, and the same

We solve d-dimensional Heesch’s problem in the asymptotic sense. Namely, we show that, if \(d\rightarrow \infty \), then there is no uniform upper bound on the set of all possible finite Heesch numbers in the space \({\mathbb {E}}^d\); in other words, given any nonnegative integer n, we can find a dimension d (depending on n) in which there exists a hypersolid whose Heesch number is finite and greater

Let \({\varvec{a}}_1,\ldots , {\varvec{a}}_r\) be vectors in a half-space of \({\mathbb {R}}^n\). We call \(C={\varvec{a}}_1{\mathbb {R}}^{+}+\cdots +{\varvec{a}}_r {\mathbb {R}}^{+}\) a convex polyhedral cone and \(\{{\varvec{a}}_1,\ldots , {\varvec{a}}_r\}\) a generator set of C. A generator set with the minimal cardinality is called a frame. We investigate the translation tilings of convex polyhedral

Problem 4.19 in Ziegler (Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)) asserts that every simple 3-dimensional polytope has the property that its dual can be constructed as the convex hull of points chosen from the facets of the original polytope. In this note we state a variant of this conjecture that requires the points to be a subset of the vertices of

We prove that it is \(\#{\mathsf {P}}\)-complete to count the triangulations of a (non-simple) polygon.

We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume m in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This

We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected \((d-1)\)-dimensional simplicial complexes on n vertices, denoted \(H_s(n, d)\). Using a probabilistic construction we give a new lower bound on \(H_s(n, d)\) that is within an \(O(d^2)\) factor of the upper bound. This improves on the previously best known lower bound which was within

We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include \(L_p\)-norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses \(O(n \log ^3

We provide a new type of proof for the Koebe–Andreev–Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph G, one can remove any flippable edge \(e^-\) of this graph and then continuously flow the disks in the plane, so that at the end of the flow, one obtains a new disk packing whose

Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms which are both simple and efficient in theory and in practice. Randomized incremental constructions are usually space-optimal and time-optimal in the worst case, as exemplified by the construction of convex

Tverberg’s theorem says that a set with sufficiently many points in \({\mathbb {R}}^d\) can always be partitioned into m parts so that the \((m-1)\)-simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results of our paper demonstrate that Tverberg’s theorem is just a special case of a more general situation, where other simplicial complexes must always arise as nerve

Given a set of n points in the plane, and a parameter \(k\), we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing \(k\) points. We present the first near quadratic time algorithm for this problem, improving over the previous near-\(O(n^{5/2})\)-time algorithm by Kaplan et al. (25th European Symposium on Algorithms. Leibniz Int Proc Inform, vol. 87, # 52

We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic measure. More generally, we also prove similar convergence

Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs

Consider a realization of a graph in the space with straight segments representing edges. Let us assign a stress for every its edge. In case if at every vertex of the graph the stresses sum up to zero, we say that the realization is a tensegrity. Some realizations possess non-zero tensegrities while the others do not. In this paper we study necessary and sufficient existence conditions for tensegrities

A lattice polytope \(\mathscr {P} \subset \mathbb {R}^d\) is called a locally anti-blocking polytope if for any closed orthant \({\mathbb R}^d_{\varepsilon }\) in \(\mathbb {R}^d\), \(\mathscr {P} \cap \mathbb {R}^d_{\varepsilon }\) is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. We give a formula for the \(h^*\)-polynomials of locally anti-blocking

We study the following separation problem: Given a collection of pairwise disjoint coloured objects in the plane with k different colours, compute a shortest “fence” F, i.e., a union of curves of minimum total length, that separates every pair of objects of different colours. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior

We consider the following setting: suppose that we are given a manifold M in \({\mathbb {R}}^d\) with positive reach. Moreover assume that we have an embedded simplical complex \({\mathcal {A}}\) without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in \({\mathcal {A}}\) have sufficient quality. We prove that if, locally, interiors of the projection

We show how to construct a \((1+\varepsilon )\)-spanner over a set \({P}\) of n points in \({\mathbb {R}}^d\) that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters \({\vartheta },\varepsilon \in (0,1)\), the computed spanner \({G}\) has $$\begin{aligned} {{\mathcal {O}}}\bigl (\varepsilon ^{-O(d)} {\vartheta }^{-6} n(\log \log n)^6 \log n \bigr ) \end{aligned}$$

Terrain visibility graphs are a well-known graph class in computational geometry. They are closely related to polygon visibility graphs, but a precise graph-theoretical characterization is still unknown. Over the last decade, terrain visibility graphs attracted considerable attention in the context of time series analysis (there called time series visibility graphs) with various practical applications

Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Fréchet distance is one of the most well-studied similarity measures. Informally, the Fréchet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we study

The famous Szemerédi–Trotter theorem states that any arrangement of n points and n lines in the plane determines \(O(n^{4/3})\) incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let \(\mathcal {L}_a\) and \(\mathcal {L}_b\) be two sets of t lines in the plane and let \(P=\{\ell _a \cap \ell _b : \ell _a \in \mathcal {L}_a, \,\ell

We present new results on a number of fundamental problems about dynamic geometric data structures: (1) We describe the first fully dynamic data structures with sublinear amortized update time for maintaining (i) the number of vertices or the volume of the convex hull of a 3D point set, (ii) the largest empty circle for a 2D point set, (iii) the Hausdorff distance between two 2D point sets, (iv) the

We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result

Richard Guy asked the following question: can we find a triangle with rational sides, medians and area? Such a triangle is called a perfect triangle and no example has been found to date. It is widely believed that such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw about rational distance sets contained in an algebraic curve, to show that for any angle \(0<\theta < \pi \)

The vertical depth relation among n pairwise openly disjoint triangles in 3-space may contain cycles. We show that, for any \(\varepsilon >0\), the triangles can be cut into \(O(n^{3/2+\varepsilon })\) connected semialgebraic pieces, whose description complexity depends only on the choice of \(\varepsilon \), such that the depth relation among these pieces is now a proper partial order. This bound

We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser

We study the number of copies of a weakly connected subdigraph of the k nearest neighbor (kNN) digraph based on data from certain random point processes in \(\mathbb {R}^d\). In particular, based on the asymptotic theory for functionals of point sets from homogeneous Poisson process (HPP) and uniform binomial process (UBP), we provide a general result for the asymptotic behavior of the number of weakly

We describe an algorithm for computing the min-convex hull of a finite collection of points in the affine building of \(\hbox {SL}_d(K)\), for K a field with discrete valuation. These min-convex hulls describe the relations among a finite collection of invertible matrices over K. As a consequence, we bound the dimension of the tropical projective space needed to realize the min-convex hull as a tropical

Convex polygons are distinguishable among the piecewise \(C^\infty \) convex domains by comparing their visual angle functions on any surrounding circle. This is a consequence of our main result, that every segment in a \(C^\infty \) multicurve can be reconstructed from the masking function of the multicurve given on any surrounding circle.

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 graph, each vertex has an opposite face with isomorphism \(\tau :G \rightarrow G^*\) (where \(G^*\) is the unique dual graph), a metric mapping is a map \(\eta :V(G) \rightarrow \mathbb

We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in \(O(n\log ^2\!n)\) time using linear space, where n is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejčič [CGTA’15]

Given a hypergraph \(\mathcal {H}=(X,{\mathcal {S}})\), a planar support for \(\mathcal {H}\) is a planar graph G with vertex set X, such that for each hyperedge \(S\in \mathcal {S}\), the subgraph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph

Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are \(344\, 843 \,867\) such cones, organized into a database of \(14\,373\,645\) symmetry classes. The Schläfli fan gives a further refinement of these cones. It reveals all possible patterns of lines on tropical cubic surfaces, thus serving as a combinatorial base space

We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point \(a \in {{\,\mathrm{{\texttt {conv}}}\,}}P\), and an integer \(r \le n\), there is a subset \(Q\subset P\) of r elements such that the distance between a and \({{\,\mathrm{{\texttt {conv}}}\,}}Q\) is less

Motivated by questions from Ehrhart theory, we present new results on discrete equidecomposability. Two rational polygons P and Q are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection (equivalently, a piecewise affine-linear bijection that preserves the integer lattice \({\mathbb {Z}}^2\)) from P to Q. We develop an invariant for a particular version of

A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair

Hypertope is a generalization of the concept of polytope, which in turn generalizes the concept of a map and hypermap, to higher rank objects. Regular hypertopes with spherical residues, which we call regular locally spherical hypertopes, must be either of spherical, euclidean, or hyperbolic type. That is, the type-preserving automorphism group of a locally spherical regular hypertope is a quotient

Let \({\mathcal {A}}=\{A_1,\ldots ,A_n\}\) be a family of sets in the plane. For \(0 \le i < n\), denote by \(f_i\) the number of subsets \(\sigma \) of \(\{1,\ldots ,n\}\) of cardinality \(i+1\) that satisfy \(\bigcap _{i \in \sigma } A_i \ne \emptyset \). Let \(k \ge 2\) be an integer. We prove that if each k-wise and \((k{+}1)\)-wise intersection of sets from \({\mathcal {A}}\) is empty, or a single

We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the

Let X be a nonempty finite subset of \({\mathbb {R}}^d\) and \(X=\bigcup _{i=1}^m X_i\) a coloring with \(m

We study symmetric non-negative forms and their relationship with symmetric sums of squares. For a fixed number of variables n and degree 2d, symmetric non-negative forms and symmetric sums of squares form closed, convex cones in the vector space of n-variate symmetric forms of degree 2d. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate

We consider the X-ray transform in a projective space over a finite field. It is well known (after Bolker) that this transform is injective. We formulate an analog of Gelfand’s admissibility problem for the Radon transform, which asks for a classification of all minimal sets of lines for which the restricted Radon transform is injective. The solution involves doubly ruled quadric surfaces.

An abstract topological graph (briefly an AT-graph) is a pair \(A=(G,{\mathcal {X}})\) where \(G=(V,E)\) is a graph and \({\mathcal {X}}\subseteq {E \atopwithdelims ()2}\) is a set of pairs of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from \({\mathcal {X}}\) crosses exactly once and no other pair crosses. We show that simply realizable