Related papers: Homotheties and incidences

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set $S$ of $s$ points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in…

In this paper we establish an improved bound for the number of incidences between a set $P$ of $m$ points and a set $H$ of $n$ planes in $\mathbb R^3$, provided that the points lie on a two-dimensional nonlinear irreducible algebraic…

We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two…

Let $M=G/K$ be a compact homogeneous space and assume that $G$ and $K$ have many simple factors. We show that the topological condition of having maximal third Betti number, in the sense that $b_3(M)=s-1$ if $G$ has $s$ simple factors, so…

Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Pl\"ucker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is \emph{non-degenerate} with respect to…

We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we…

In the setting of spaces of homogeneous type, we study some Hardy type inequalities, which notably appeared in the proofs of local T(b) theorems as in [AR]. We give some suffi cient conditions ensuring their validity, related to the…

This paper is devoted to the study of the $m$-point homogeneity property for the vertex sets of polytopes in Euclidean spaces. In particular, we present the classifications of $2$-point and $3$-point homogeneous polyhedra in $\mathbb{R}^3$.

We give a fairly elementary and simple proof that shows that the number of incidences between $m$ points and $n$ lines in ${\mathbb R}^3$, so that no plane contains more than $s$ lines, is $$ O\left(m^{1/2}n^{3/4}+ m^{2/3}n^{1/3}s^{1/3} + m…

We present simple examples of finite-dimensional connected homogeneous spaces (they are actually topological manifolds) with nonhomogeneous and nonrigid factors. In particular, we give an elementary solution of an old problem in general…

We develop new tools to compute the index of symmetry in the context of homogeneous fibrations. As a consequence of our results, we determine the index of symmetry of every homogeneous space diffeomorphic to a compact rank-one symmetric…

The homotheties of spherically symmetric spacetimes admitting $G_4$ , $G_6$ and $G_{10}$ as maximal isometry groups are already known, whereas for the space-times admitting $G_3$ as isometry groups, the solution in the form of differential…

Discrete point sets $\mathcal{S}$ such as lattices or quasiperiodic Delone sets may permit, beyond their symmetries, certain isometries $R$ such that $\mathcal{S}\cap R\mathcal{S}$ is a subset of $\mathcal{S}$ of finite density. These are…

In the point set embeddability problem, we are given a plane graph $G$ with $n$ vertices and a point set $S$ with $n$ points. Now the goal is to answer the question whether there exists a straight-line drawing of $G$ such that each vertex…

We study a wide spectrum of incidence problems involving points and curves or points and surfaces in $\mathbb R^3$. The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [2010,2015], requires a…

We prove almost tight bounds on incidences between points and $k$-dimensional varieties of bounded degree in $\R^d$. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points.

A brief summary of results on homotheties in General Relativity is given, including general information about space-times admitting an r-parameter group of homothetic transformations for r>2, as well as some specific results on perfect…

The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the…

We study homogeneity aspects of metric spaces in which all triples of distinct points admit pairwise different distances; such spaces are called isosceles-free. In particular, we characterize all homogeneous isosceles-free spaces up to…

We study in this paper three natural notions of convergence of homogeneous manifolds, namely infinitesimal, local and pointed, and their relationship with a fourth one, which only takes into account the underlying algebraic structure of the…