Related papers: A refined energy bound for perpendicular bisectors

Poncelet's theorem states that if there exists an n-sided polygon which is inscribed in a given conic C and circumscribed about another conic D, then there are infinitely many such n-gons. Proofs of this theorem that we are aware of,…

Let $L_{a,b}$ be a line in the Euclidean plane with slope $a$ and intercept $b$. The dimension spectrum $\spec(L_{a,b})$ is the set of all effective dimensions of individual points on $L_{a,b}$. The dimension spectrum conjecture states…

We estimate locations of the regions of the percolation and of the non-percolation in the plane $(\lambda,\beta)$: the Poisson rate -- the inverse temperature, for interacted particle systems in finite dimension Euclidean spaces. Our…

In 1954, G. Polya conjectured that the counting function $N(\Omega,\Lambda)$ of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set $\Omega\subset R^d$ is lesser (resp. greater)…

Let $Q$ be a finite set of points in the plane. For any set $P$ of points in the plane, $S_{Q}(P)$ denotes the number of similar copies of $Q$ contained in $P$. For a fixed $n$, Erd\H{o}s and Purdy asked to determine the maximum possible…

We prove that every set of $n$ red and $n$ blue points in the plane contains a red and a blue point such that every circle through them encloses at least $n(1-\frac{1}{\sqrt{2}}) -o(n)$ points of the set. This is a two-colored version of a…

In 2016, a breakthrough result of Chechik and Wulff-Nilsen [SODA '16] established that every $n$-node graph $G$ has a $(1+\varepsilon)(2k-1)$-spanner of lightness $O_{\varepsilon}(n^{1/k})$, and recent followup work by Le and Solomon [STOC…

Given a finite point set $P\subset\mathbb{R}^d$, we call a multiset $A$ a one-sided weak $\varepsilon$-approximant for $P$ (with respect to convex sets), if $|P\cap C|/|P|-|A\cap C|/|A|\leq\varepsilon$ for every convex set $C$. We show…

We confirm the following conjecture of Fekete and Woeginger from 1997: for any sufficiently large even number $n$, every set of $n$ points in the plane can be connected by a spanning tour (Hamiltonian cycle) consisting of straight-line…

Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries…

Let alpha be an arbitrary angle in a random spherical triangle Delta and a be the side opposite alpha. (The sphere has radius 1; vertices of Delta are independent and uniform.) If some other side is constrained to be pi/2, then…

We give a short proof that for a bounded domain $\Omega\subset\mathbb{R}^n$ and continuous boundary data $g\in C(\partial\Omega)$ admitting a continuous finite-energy extension $\phi\in H^{1}(\Omega)\cap C(\bar\Omega)$, the minimizer of the…

Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$…

New condition is found for the set of points in the plane, for which the locus is a circle. It is proved: the locus of points, such that the sum of the $(2m)$-th powers $S_n^{(2m)}$}of the distances to the vertexes of fixed regular…

Given $\alpha \in (0, \infty)$ and $r \in (0, \infty)$, let ${\cal D}_{r, \alpha}$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha^2$. Consider the Poisson point process having uniform intensity density on ${\cal…

A double-normal pair of a finite set $S$ of points from Euclidean space is a pair of points $\{p,q\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $p$ and $q$ that are perpendicular to $pq$. A…

An arrangement of circles in which circles intersect only in angles of $\pi/2$ is called an \emph{arrangement of orthogonal circles}. We show that in the case that no two circles are nested, the intersection graph of such an arrangement is…

We derive an analytical trace formula for the level density of the two-dimensional elliptic billiard using an improved stationary phase method. The result is a continuous function of the deformation parameter (eccentricity) through all…

A hyperbolic set on a compact manifold M, satisfies the property: given two of your any points p and q, such that for all positive \epsilon>0, there is a trajectory in the hyperbolic set from a point \epsilon-close to p to a point…

We consider a question raised by Rudnev: given four pencils of $n$ concurrent lines in $\mathbb R^2$, with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main…