Related papers: Some results about the Tight Span of spheres
The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has…
We prove that a nonempty closed and geodesically convex subset of the $l_{\infty}$ plane $\mathbb{R}^2_{\infty}$ is hyperconvex and we characterize the tight spans of arbitrary subsets of $\mathbb{R}^2_{\infty}$ via this property: Given any…
Given a finite metric, one can construct its tight span, a geometric object representing the metric. The dimension of a tight span encodes, among other things, the size of the space of explanatory trees for that metric; for instance, if the…
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…
A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a…
We use the trimming transformations to study the tight span of a metric space.
The tight span $T_d$ of a metric $d$ on a finite set is the subcomplex of bounded faces of an unbounded polyhedron defined by~$d$. If $d$ is generic then $T_d$ is known to be dual to a regular triangulation of a second hypersimplex. A tight…
The aim of this paper is to present some properties of reduced spherical convex bodies on the two-dimensional sphere $S^2$. The intersection of two different non-opposite hemispheres is called a lune. By its thickness we mean the distance…
Let $M$ be a complete Riemannian $3$-manifold with sectional curvatures between $0$ and $1$. A minimal $2$-sphere immersed in $M$ has area at least $4\pi$. If an embedded minimal sphere has area $4\pi$, then $M$ is isometric to the unit…
A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\Phi:X\to exp(X)$ such that $[x,y]\subset\Phi(x)\cup \Phi(y)$ for all $x,y\in X$. We prove that each convex subset of…
A thin tube is an $n$-dimensional space which is very thin in $n-1$ directions, compared to the remaining direction, for example the $\epsilon$-neighborhood of a curve or an embedded graph in $\R^n$ for small $\epsilon$. The Laplacian on…
Metric embeddings are central to metric theory and its applications. Here we consider embeddings of a different sort: maps from a set to subsets of a metric space so that distances between points are approximated by minimal distances…
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
We study the shortest geodesics on flat cone spheres, i.e. flat metrics on the sphere with conical singularities. The length of the shortest geodesic between two singular points can be treated as a function on the moduli space of flat cone…
What one obtains when the min-max methods for the distance function are applied on the space of pairs of points of a Riemannian two-sphere? This question is studied in details in the present article. We show that the associated min-max…
We present a survey article about the geometry of convex bodies on the $d$-dimensional sphere $S^d$. We concentrate on the results based on the notion of the width of a convex body $C \subset S^d$ determined by a supporting hemisphere of…
V. Turaev defined recently an operation of "Trimming" for pseudo-metric spaces and analysed the tight span of (pseudo-)metric spaces via this process. In this work we investigate the trimming of finite subspaces of the Manhattan plane. We…
This thesis consists of five papers about reduced spherical convex bodies and in particular spherical bodies of constant width on the $d$-dimensional sphere $S^d$. In paper I we present some facts describing the shape of reduced bodies of…
In the recent work [Metrically round and sleek metric spaces, \emph{The Journal of Analysis} (2022), pp 1--17], the authors proved some results on metrically round and sleek linear metric spaces and metric spaces. In continuation, the…