Related papers: Extra-large metrics

For a family $\mathcal{C}$ of properly embedded curves in the 2-dimensional disk $\mathbb{D}^{2}$ satisfying certain uniqueness properties, we consider convex polygons $P\subset \mathbb{D}^{2}$ and define a metric $d$ on $P$ such that…

Recently, the first author together with Jens Marklof studied generalizations of the classical three distance theorem to higher dimensional toral rotations, giving upper bounds in all dimensions for the corresponding numbers of distances…

We describe, in terms of generalized elliptic integrals, the hyperbolic metric of the twice-punctured sphere with one conical singularity of prescribed order. We also give several monotonicity properties of the metric and a couple of…

We prove a theorem on the relationships between the lengths of sides of a spherical quadrilateral with three right angles. They are analogous to the relationships in the Lambert quadrilateral in the hyperbolic plane. We apply this theorem…

Let $\Omega$ be a domain in $\mathbb{C}$ with hyperbolic metric $\lambda_\Omega(z)|dz|$ of Gaussian curvature $-4.$ Mejia and Minda proved in their 1990 paper that $\Omega$ is (Euclidean) convex if and only if…

Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but…

Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces,…

An $F$-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated…

We use a constrained Monte Carlo technique to analyze ultrametric features of a 4 dimensional Edwards-Anderson spin glass with quenched couplings J=\pm 1. We find that in the large volume limit an ultrametric structure emerges quite clearly…

Shape metrics for objects in high dimensions remain sparse. Those that do exist, such as hyper-volume, remain limited to objects that are better understood such as Platonic solids and $n$-Cubes. Further, understanding objects of ill-defined…

We first introduce a configuration of arbitrary isogonal conjugates related to a known property concerning the spiral center of two pairs of isogonal conjugates. We then consider a special case where two conics are tangent at exactly two…

Euler's inequality is a well known inequality relating the inradius and circumradius of a triangle. In Euclidean geometry, this inequality takes the form $R \geq 2r$ where $R$ is the circumradius and $r$ is the inradius. In spherical…

We give a variety of uniqueness results for minimal ellipsoids circumscribing and maximal ellipsoids inscribed into a convex body. Uniqueness follows from a convexity or concavity criterion on the function used to measure the size of the…

We study a wide class of metrics in a Lebesgue space with a standard measure, the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the…

We classify all Kahler metrics in an open subset of $C^2$ whose real geodesics are circles. All such metrics are equivalent (via complex projective transformations) to Fubini metrics (i.e. to Fubini-Study metric on $CP^2$ restricted to an…

Symmetric cones can be endowed with at least two interesting non Riemannian metrics: the Hilbert and the Thompson metrics. It is trivial that the linear maps preserving the cone are isometries for those two metrics. Oddly enough those are…

An $L$-spherical code is a set of Euclidean unit vectors whose pairwise inner products belong to the set $L$. We show, for a fixed $\alpha,\beta>0$, that the size of any $[-1,-\beta]\cup\{\alpha\}$-spherical code is at most linear in the…

In this article, we consider linearly convex complex cones in complex Banach spaces and we define a new projective metric on these cones. Compared to the hyperbolic gauge of Rugh, it has the advantage of being explicit, and easier to…

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric $g$ is geodesically rigid: if any other complete metric $\bar g$ has the same (unparametrized) geodesics with $g$, then the Levi-Civita connections of $g$ and…

For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic…