In 1984, Dancis proved that any d-dimensional simplicial manifold is determined by its \((\lfloor d/2\rfloor +1)\)-skeleton. This paper adapts his proof to the setting of cubical complexes that can be embedded into a cube of arbitrary dimension. Under some additional conditions (for example, if the cubical manifold is a sphere), the result can be tightened to the \(\lceil {d}/2\rceil \)-skeleton when

We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent definitions of the facial weak order of a central arrangement: (1) by exploiting the fact that the faces are intervals in the poset of regions, (2) by describing its cover

We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Möbius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence

A polytope \({{ \mathsf {P}}}\) in some euclidean space is called quasi-regular if each facet \({{ \mathsf {F}}}\) of \({{ \mathsf {P}}}\) is regular and the symmetry group \({\mathbf {G}}({{ \mathsf {F}}})\) of \({{ \mathsf {F}}}\) is a subgroup of the symmetry group \({\mathbf {G}}({{ \mathsf {P}}})\) of \({{ \mathsf {P}}}\). Further, \({{ \mathsf {P}}}\) is of full rank if its rank and dimension

We study the problem of maximizing the minimal value over the sphere \(S^{d-1}\subset {\mathbb {R}}^d\) of the potential generated by a configuration of \(d+1\) points on \(S^{d-1}\) (the maximal discrete polarization problem). The points interact via the potential given by a function f of the Euclidean distance squared, where \(f:[0,4]\rightarrow (-\infty ,\infty ]\) is continuous (in the extended

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of \(S^3\) branched along a knot \(\alpha \subset S^3\). Since every closed, oriented three-manifold admits such a presentation

The typical cell of a Voronoi tessellation generated by \(n+1\) uniformly distributed random points on the d-dimensional unit sphere \(\mathbb {S}^d\) is studied. Its f-vector is identified in distribution with the f-vector of a beta’ polytope generated by n random points in \(\mathbb {R}^d\). Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases \(d\in \{2

Using the ordered analogue of Farley–Sabalka’s discrete gradient field on the configuration space of a graph, we unravel a levelwise behavior of the generators of the pure braid group on a tree. This allows us to generalize Farber’s equivariant description of the homotopy type of the configuration space on a tree on two particles. The results are applied to the calculation of all the higher topological

A famous and wide-open problem, going back to at least the early 1970s, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute such inequalities when a chromatic polynomial \(\chi _G(n)=\chi ^*_0\left( {\begin{array}{c}n+d\\ d\end{array}}\right) +\chi

We provide a formula for the Ehrhart polynomial of the connected matroid of size n and rank k with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and \(h^*\)-real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart

The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard (Ann. Statist. 47(1), 177–204 (2019)), an estimator

We study planar polygonal curves with the variational methods. We show a unified interpretation of discrete curvatures and the Steiner-type formula by extracting the notion of the discrete curvature vector from the first variation of the length functional. Moreover, we determine the equilibrium curves for the length functional under the area-constraint condition and study their stability.

Gardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown in Sidiropoulos and Sridhar (33rd International Symposium on Computational Geometry (Brisbane 2017). Leibniz Int. Proc. Inform., vol. 77, # 58. Leibniz-Zent. Inform., Wadern, 2017) that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension

A metric graph is a pair (G, d), where G is a graph and \(d:E(G) \rightarrow \mathbb {R}_{\ge 0}\) is a distance function. Let \(p \in [1,\infty ]\) be fixed. An isometric embedding of the metric graph (G, d) in \(\ell _p^k = (\mathbb {R}^k, d_p)\) is a map \(\phi :V(G) \rightarrow \mathbb {R}^k\) such that \(d_p(\phi (v), \phi (w)) = d(vw)\) for all edges \(vw\in E(G)\). The \(\ell _p\)-dimension

We give a new algorithm to simplify a given triangulation with respect to a given curve. The simplification uses flips together with powers of Dehn twists in order to complete in polynomial time in the bit-size of the curve.

We give a short solution of the kissing number problem in dimension three.

We describe a uniform approach to two known graph drawing results including Gioan’s theorem, stating that any two good drawings of a complete graph with the same rotation system are isomorphic up to Reidemeister moves of type 3, and a characterization of pseudolinear drawings of the complete graph via an excluded configuration: a bad \(K_4\). Our approach yields a new and short self-contained proof

In Beltrán and Etayo (J. Complexity 59, # 101471 (2020)) the authors presented a family of points on the sphere \(\mathbb {S}^{2}\) depending on many parameters, called the Diamond ensemble. In this paper we compute the spherical cap discrepancy of the Diamond ensemble as well as some other quantities. We also define an area regular partition on the sphere where each region contains exactly one point

The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for n-dimensional rectangle decomposable persistence modules up to a constant \(2n-1\) that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved

We say that a triangle T tiles a polygon A, if A can be dissected into finitely many nonoverlapping triangles similar to T. We show that if \(N>42\), then there are at most three nonsimilar triangles T such that the angles of T are rational multiples of \(\pi \) and T tiles the regular N-gon. A tiling into similar triangles is called regular, if the pieces have two angles, \(\alpha \) and \(\beta \)

We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a \(\Gamma \)-permutahedron for some finite reflection group \(\Gamma \subset {{\,\mathrm{O}\,}}(\mathbb {R}^d)\). The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length. The classification of these then follows from the classification

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 representation of certain gain graphs. As

This paper studies visibility problems in Euclidean spaces $\mathbb{R}^d$ where the obstacles are the points of infinite discrete sets $Y\subseteq\mathbb{R}^d$. A point $x\in\mathbb{R}^d$ is called $\varepsilon$-visible for $Y$ (notation: $x\in\mathbf{vis}(Y, \varepsilon))$ if there exists a ray $L\subseteq\mathbb{R}^d$ emanating from $x$ such that $||y-z||\geq\varepsilon$, for all $y\in Y\setminus\{x\}$

In this project we show the existence of arbitrary length arithmetic progressions in model sets and Meyer sets in the Euclidean $d$-space. We prove a van der Waerden type theorem for Meyer sets. We show that pure point 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

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