Related papers: Geodesic paths in the finite dimensional unit sphe…
In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy…
For sufficiently tame paths in $\mathbb{R}^n$, Euclidean length provides a canonical parametrization of a path by length. In this paper we provide such a parametrization for all continuous paths. This parametrization is based on an…
We provide a self-contained geometric description of the geodesic flow in the three-dimensional Lie group $\mathrm{Sol}$, one of Thurston's eight model geometries. The geometry of geodesics is governed by a single invariant $k\in[0,1]$, its…
It is an open problem to show that in two-dimensional first-passage percolation, the sequence of finite geodesics from any point to $(n,0)$ has a limit in $n$. In this paper, we consider this question for first-passage percolation on a wide…
A set of points $S$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ is called a 2-distance set if the set of pairwise distances between the points has cardinality two. The 2-distance set is called spherical if its points lie on the unit…
We perform an experimental evaluation of algorithms for finding geodesic shortest paths between two points inside a simple polygon in the constant-workspace model. In this model, the input resides in a read-only array that can be accessed…
We study the behaviour of quasi-geodesics in Out(F_n). Given an element f in Out(F_n) there are several natural paths connecting the origin to f in Out(F_n); for example, paths associated to sequences of Stallings folds and paths induced by…
The symmetries of paths in a manifold $M$ are classified with respect to a given pointwise proper action of a Lie group $G$ on $M$. Here, paths are embeddings of a compact interval into $M$. There are at least two types of symmetries:…
An arrangement of pseudocircles is a finite set of oriented closed Jordan curves each two of which cross each other in exactly two points. To describe the combinatorial structure of arrangements on closed orientable surfaces, in (Linhart,…
We prove a bound for the geodesic diameter of a subset of the unit ball in $\mathbb{R}^n$ described by a fixed number of quadratic equations and inequalities, which is polynomial in $n$, whereas the known bound for general degree is…
We endow the set of probability measures on a weighted graph with a Monge--Kantorovich metric, induced by a function defined on the set of vertices. The graph is assumed to have $n$ vertices and so, the boundary of the probability simplex…
We study geodesics of Hofer's metric on the space of Lagrangian submanifolds in arbitrary symplectic manifolds from the variational point of view. We give a characterization of length-critical paths with respect to this metric. As a result,…
Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit metric. We study the…
We define the notion of near geodesic between points of a metric space when no geodesic exists, and use this to extend Recio-Mitter's notion of geodesic complexity to non-geodesic spaces. This has potential application to topological…
The goal of this paper is to study the path-following method for univariate polynomials. We propose to study the complexity and condition properties when the Newton method is applied as a correction operator. Then we study the geodesics and…
We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on $Z$-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a…
Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of…
We consider the space of probability measures on a discrete set $X$, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $Y \subseteq X$, it is natural to ask whether they can be connected…
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 construct all Finsler metrics on the two-sphere for which geodesics are circles and show that any (reversible) path geometry on a two-dimensional manifold is locally the system of geodesics of a Finsler metric.