We characterise finite and infinitesimal rigidity for bar-joint frameworks in \({\mathbb {R}}^d\) with respect to polyhedral norms (i.e. norms with closed unit ball \({\mathcal {P}}\), a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in \({\mathbb {R}}^d\) which are well-positioned with respect to \({\mathcal...

We study the monoid generated by \(n \times n\) distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \), and we compute the structure of this fan for \(n\) up to \(5\). The monoid captures gossip among...

Recently, it was proved that triangle-free intersection graphs of \(n\) line segments in the plane can have chromatic number as large as \(\Theta (\log \log n)\). Essentially the same construction produces \(\Theta (\log \log n)\)-chromatic triangle-free intersection graphs of a variety of other geometric shapes—those belonging to any class of compact arc-connected sets in...

It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.

This paper extends the scenario of the Four Color Theorem in the following way. Let \(\fancyscript{H}_{d,k}\) be the set of all \(k\)-uniform hypergraphs that can be (linearly) embedded into \(\mathbb {R}^d\). We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in \(\fancyscript{H}_{d,k}\). For example, we can prove that for \(d\ge 3...

One classical result of Freiman gives the optimal lower bound for the cardinality of \(A+A\) if \(A\) is a \(d\)-dimensional finite set in \(\mathbb R^d\). Matolcsi and Ruzsa have recently generalized this lower bound to \(|A+kB|\) if \(B\) is \(d\)-dimensional, and \(A\) is contained in the convex hull of \(B\). We characterize the equality case of the Matolcsi–Ruzsa bound. The...

We study constructions of \(k \times n\) matrices \(A\) that both (1) satisfy the restricted isometry property (RIP) at sparsity \(s\) with optimal parameters, and (2) are efficient in the sense that only \(O(n\log n)\) operations are required to compute \(Ax\) given a vector \(x\). Our construction is based on repeated application of independent transformations of the form \(DH...

A family of sets in the plane is simple if the intersection of any subfamily is arc-connected, and it is pierced by a line \(L\) if the intersection of any member with \(L\) is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their...

A symmetric \(n\)-Venn diagram is one that is invariant under \(n\)-fold rotation, up to a relabeling of curves. A simple \(n\)-Venn diagram is an \(n\)-Venn diagram in which at most two curves intersect at any point. In this paper, we introduce a new property of Venn diagrams called crosscut symmetry, which is related to dihedral symmetry. Utilizing a computer search restricted...

We investigate the homotopy type of the Alexander dual of a simplicial complex. It is known that in general the homotopy type of \(K\) does not determine the homotopy type of its dual \(K^*\). We construct for each finitely presented group \(G\), a simply connected simplicial complex \(K\) such that \(\pi _1(K^*)=G\) and study sufficient conditions on \(K\) for \(K^*\) to have...

In this paper, we answer Larman’s question on Borsuk’s conjecture for two-distance sets. We find a two-distance set consisting of 416 points on the unit sphere \(S^{64}\subset\mathbb{R}^{65}\) which cannot be partitioned into 83 parts of smaller diameter. This also reduces the smallest dimension in which Borsuk’s conjecture is known to be false. Other examples of two-distance...

We present a new probabilistic symbolic algorithm that, given a variety defined in an n-dimensional affine space by a generic sparse system with fixed supports, computes the Zariski closure of its projection to an ℓ-dimensional coordinate affine space with ℓ<n. The complexity of the algorithm depends polynomially on some combinatorial invariants associated to the supports.

Realisations of associahedra with linear non-isomorphic normal fans can be obtained by alteration of the right-hand sides of the facet-defining inequalities from a classical permutahedron. These polytopes can be expressed as Minkowski sums and differences of dilated faces of a standard simplex as described by Ardila et al. (Discret Comput Geom, 43:841–854, 2010). The coefficients...

Given a combinatorial triangulation of an n-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for the areas of the triangles in such drawings. We define a generalized notion of triangulation, and we show that the areas of the triangles in a...

In this paper we present a new technique to construct neighborly polytopes, and use it to prove a lower bound of \({\big (( r+d ) ^{( \frac{r}{2}+\frac{d}{2} )^{2}}\big )}\big /{\big ({r}^{{(\frac{r}{2})}^{2}} {d}^{{(\frac{d}{2})}^{2}}{\mathrm{e}^{3\frac{r}{2}\frac{d}{2}}}\big )}\) for the number of combinatorial types of vertex-labeled neighborly polytopes in even dimension d...

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set \(X\) in \(\mathbb{R }^2\) that is not an...

Cube tilings formed by \(n\)-dimensional \(4\mathbb Z ^n\)-periodic hypercubes with side \(2\) and integer coordinates are considered here. By representing the problem of finding such cube tilings within the framework of exact cover and using canonical augmentation, pairwise nonisomorphic 5-dimensional cube tilings are exhaustively enumerated in a constructive manner. There are...

We consider the problem of reconstructing a compact 3-manifold (with boundary) embedded in \(\mathbb R ^3\) from its cross-sections \(\mathcal{S }\) with a given set of cutting planes \(\mathcal P \) having arbitrary orientations. In this paper, we analyse a very natural reconstruction strategy: a point \(x \in \mathbb R ^3\) belongs to the reconstructed object if (at least one...

A simple topological graph \(T=(V(T), E(T))\) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs \(G\) and \(H\) are isomorphic if \(H\) can be obtained from \(G\) by a homeomorphism of the sphere, and weakly isomorphic if \(G\) and \(H\) have...

We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \(\chi \) on the set of 3-subsets of a finite set \(I\) is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset \(J\) of \(I\) the...

For every \(k>3\), we give a construction of planar point sets with many collinear \(k\)-tuples and no collinear \((k+1)\)-tuples. We show that there are \(n_0=n_0(k)\) and \(c=c(k)\) such that if \(n\ge n_0\), then there exists a set of \(n\) points in the plane that does not contain \(k+1\) points on a line, but it contains at least \(n^{2-({c}/{\sqrt{\log n}})}\) collinear \(k...