Related papers: Egan conjecture holds

For a set P of n points in R^2, the Euclidean 2-center problem computes a pair of congruent disks of the minimal radius that cover P. We extend this to the (2,k)-center problem where we compute the minimal radius pair of congruent disks to…

We study metric spheres Z obtained by gluing two hemispheres of the Euclidean sphere along an orientation-preserving homeomorphism mapping the equator onto itself, where the distance on Z is the canonical distance that is locally isometric…

A submanifold $M \subset R^N$ is $r$-neighborly if for any $r$ points in $M$ there is a hyperplane, supporting $M$ and touching it at exactly these $r$ points. We prove that the minimal dimension $\Delta(k,r)$ of the Euclidean space,…

Let F be a commutative field of characteristic 0, G_n: F^n \times F^n -> F, G_n((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say that g:R^n->F^n preserves distance d>=0 if for each x,y \in R^n |x-y|=d implies…

We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane is either very sparse or has all but an…

We discuss the concept of the shadow boundary of a centrally symmetric convex ball $K$ (actually being the unit ball of a Minkowski normed space) with respect to a direction ${\bf x}$ of the Euclidean n-space $R^n$. We introduce the concept…

In 1960, Klee showed that a subset of a Euclidean space must be a singleton provided that each point in the space has a unique farthest point in the set. This classical result has received much attention; in fact, the Hilbert space version…

Let $f\colon M^{2n}\to\R^{2n+p}$, $2\leq p\leq n-1$, be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng conjectured in \cite{YZ} that if the codimension is $p\leq 11$ then, along any connected component of…

We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an extremely…

This paper considers an extremal version of the Erd\H{o}s distinct distances problem. For a point set $P \subset \mathbb R^d$, let $\Delta(P)$ denote the set of all Euclidean distances determined by $P$. Our main result is the following: if…

We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap $\mathbb{L}_2$-discrepancy of any $N$ points on the unit sphere $\mathbb S^d$ in $\mathbb{R}^{d+1}$,…

We use that the $n$-sphere for $n\geq 2$ is simply-connected to prove the Poincar\'e-Birkhoff-Witt Theorem.

A subset of the sphere is said short if it is contained in an open hemisphere. A short closed set which is geodesically convex is called a cap. The following theorem holds: 1. The minimal number of short closed sets covering the $n$-sphere…

We prove that if the Hausdorff dimension of $E\subset\mathbb{R}^d$, $d\geq 2$ is greater than $\frac{d}{2}+\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $$\Delta_2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset\mathbb{R}^2$$ has…

Given a regular $n$-gon on the plane, it is evident that from any point on the plane, taken as a center, one can draw $n$ concentric circles such that each circle passes through one of the vertices of the polygon. Naturally, this raises the…

Let $ES_{d}(n)$ be the smallest integer such that any set of $ES_{d}(n)$ points in $\mathbb{R}^{d}$ in general position contains $n$ points in convex position. In 1960, Erd\H{o}s and Szekeres showed that $ES_{2}(n) \geq 2^{n-2} + 1$ holds,…

We show that a realization of a closed connected PL-manifold of dimension n-1 in Euclidean n-space (n>2) is the boundary of a convex polyhedron if and only if the interior of each (n-3)-face has a point, which has a neighborhood lying on…

In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with…

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be…

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded…