Let \({\mathscr {S}}\) be a set of n points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of \({\mathscr {S}}\) is less than \(Kn^3\) for some \(K=o(n^{{1}/{7}})\) then, for n sufficiently large, all but at most O(K) points of \({\mathscr {S}}\) are contained in the intersection of five linearly

Suppose that n is not a prime power and not twice a prime power. We prove that for any Hausdorff compactum X with a free action of the symmetric group \({\mathfrak {S}}_n\), there exists an \({\mathfrak {S}}_n\)-equivariant map \(X \rightarrow {{\mathbb {R}}}^n\) whose image avoids the diagonal \(\{(x,x,\dots ,x)\in {{\mathbb {R}}}^n\mid x\in {{\mathbb {R}}}\}\). Previously, the special cases of this

We complete the proof of the upper bound \({\hat{\rho }}_3\le 10R\) for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all \(10R\)-clusters of a Delone set X in \({\mathbb {R}}^3\) with parameters (r, R) are equivalent, then X is regular (has a transitive symmetry group).

To embed the bouquet of g circles \(B_g\) into the n-sphere \(S^n\) so that its full symmetry group action extends to an orthogonal actions on \(S^n\), the minimal n is \(2g-1\). This answers a question raised by Zimmermann.

A surface with boundary is randomly generated by gluing polygons along some of their sides. We show that its genus and number of boundary components asymptotically follow a bivariate normal distribution.

Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particular, we show that if a set P of n points is in convex position, then the largest intersecting family of triangles with vertices in P contains at most \(

Inspired by the seminal works of Khuller et al. (SIAM J. Comput. 25(2), 355–368 (1996)) and Chan (Discrete Comput. Geom. 32(2), 177–194 (2004)) we study the bottleneck version of the Euclidean bounded-degree spanning tree problem. A bottleneck spanning tree is a spanning tree whose largest edge-length is minimum, and a bottleneck degree-K spanning tree is a degree-K spanning tree whose largest edge-length

This paper studies the problem of recovering a signal from one-bit compressed sensing measurements under a manifold model; that is, assuming that the signal lies on or near a manifold of low intrinsic dimension. We provide a convex recovery method based on the Geometric Multi-Resolution Analysis and prove recovery guarantees with a near-optimal scaling in the intrinsic manifold dimension. Our method

Let \(T=\{\triangle _1,\ldots ,\triangle _n\}\) be a set of n triangles in \(\mathbb {R}^3\) with pairwise-disjoint interiors, and let B be a convex polytope in \(\mathbb {R}^3\) with a constant number of faces. For each i, let \(C_i = \triangle _i \oplus r_i B\) denote the Minkowski sum of \(\triangle _i\) with a copy of B scaled by \(r_i>0\). We show that if the scaling factors \(r_1, \ldots , r_n\)

Eggleston (Approximation to plane convex curves. I. Dowker-type theorems. Proc. Lond. Math. Soc. 7, 351–377 (1957)) proved that in the Euclidean plane the best approximating convex n-gon to a convex disc K is always inscribed in K if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston’s statement holds in the hyperbolic plane, and we give an example showing that

Line systems passing through the origin of the d-dimensional Euclidean space admitting exactly two distinct angles are called biangular. It is shown that the maximum cardinality of biangular lines is at least \(2(d-1)(d-2)\), and this result is sharp for \(d\in \{4,5,6\}\). Connections to binary codes, few-distance sets, and association schemes are explored, along with their multiangular generalization

A seminal theorem of Tverberg states that any set of \(T(r,d)=(r-1)(d+1)+1\) points in \({\mathbb {R}}^{d}\) can be partitioned into r subsets whose convex hulls have non-empty r-fold intersection. Almost any collection of fewer points in \({\mathbb {R}}^{d}\) cannot be so divided, and in these cases we ask if the set can nonetheless be P(r, d)-partitioned, i.e., split into r subsets so that there

Given a locally finite \(X \subseteq {{{\mathbb {R}}}}^d\) and a radius \(r \ge 0\), the k-fold cover of X and r consists of all points in \({{{\mathbb {R}}}}^d\) that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both

Due to a Production error the original online publication of this article was not Open Access. The original article has been revised and is now Open Access.

The goal of this paper is to design a simplex algorithm for linear programs on lattice polytopes that traces “short” simplex paths from any given vertex to an optimal one. We consider a lattice polytope P contained in \([0,k]^n\) and defined via m linear inequalities. Our first contribution is a simplex algorithm that reaches an optimal vertex by tracing a path along the edges of P of length in \(O(n^{4}

The Baldoni–Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function, fundamental in representation theory. On the other hand, the Ehrhart polynomials can be read off from the volume functions of flow polytopes.

We prove that a set A of at most q non-collinear points in the finite plane \(\mathbb {F}_{q}^{2}\) spans more than \({|A|}/\!{\sqrt{q}}\) directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing

We show that any two geometric triangulations of a closed hyperbolic, spherical, or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes, and bound on the lengths of edges of the triangulation. This leads to an algorithm to check from the combinatorics

A string graph is the intersection graph of curves in the plane. We prove that for every \(\epsilon >0\), if G is a string graph with n vertices such that the edge density of G is below \({1}/{4}-\epsilon \), then V(G) contains two linear sized subsets A and B with no edges between them. The constant 1/4 is a sharp threshold for this phenomenon as there are string graphs with edge density less than

A d-dimensional framework is a pair (G, p), where \(G=(V,E)\) is a graph and p is a map from V to \(\mathbb {R}^d\). The length of an edge \(uv\in E\) in (G, p) is the distance between p(u) and p(v). The framework is said to be globally rigid in \(\mathbb {R}^d\) if every other d-dimensional framework (G, q), in which the corresponding edge lengths are the same, is congruent to (G, p). In a recent

Shellings of simplicial complexes have long been a useful tool in topological and algebraic combinatorics. Shellings of a complex expose a large amount of information in a helpful way, but are not easy to construct, often requiring deep information about the structure of the complex. It is natural to ask whether shellings may be efficiently found computationally. In a recent paper, Goaoc, Paták, Patáková

For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in \(L^p\), \(p\ge e(H)\), denoted by t(H, W). One may then define corresponding functionals \(\Vert W\Vert _{H}\,{:}{=}\,|t(H,W)|^{1/e(H)}\) and \(\Vert W\Vert _{r(H)}\,{:}{=}\,t(H,|W|)^{1/e(H)}\), and say that H is (semi-)norming if \(\Vert \,{\cdot }\,\Vert _{H}\) is a (semi-)norm

We prove an upper bound of the form \(2^{O(d^2 \mathop {\mathrm {polylog}}d)}\) on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones, and d-configurations. This in particular answers positively a conjecture of Bohn et al. on 2-level polytopes. We obtain our upper bound by relating affine (resp. linear) equivalence classes of

In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets.

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