# Compact sets and the closure of their convex hulls in CAT(0) spaces

@inproceedings{Berdellima2021CompactSA, title={Compact sets and the closure of their convex hulls in CAT(0) spaces}, author={Arian Berdellima}, year={2021} }

We study the closure of the convex hull of a compact set in a complete CAT(0) space. First we give characterization results in terms of compact sets and the closure of their convex hulls for locally compact CAT(0) spaces that are either regular or satisfy the geodesic extension property. Later inspired by a geometric interpretation of Carathéodory’s Theorem we introduce the operation of threading for a given set. We show that threading exhibits certain monotonicity properties with respect to… Expand

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SHOWING 1-10 OF 19 REFERENCES

A Geometric Study of the Wasserstein Space of the Line

- 2009

The concept of optimal transportation raised recently a growing interest in link with the geometry of metric spaces. In particular the L Wasserstein space W (X) have been used in [6] and [8, 9] to… Expand

Shortest Paths and Convex Hulls in 2D Complexes with Non-Positive Curvature

- Computer Science, Mathematics
- Comput. Geom.
- 2020

The use of shortest path maps to answer single-source shortest path queries in 2-dimensional CAT(0) polyhedral complexes is explored, and efficient solutions for 2-manifold and rectangular cases are unify. Expand

Lectures on Spaces of Nonpositive Curvature

- Mathematics
- 1995

I. On the interior geometry of metric spaces.- 1. Preliminaries.- 2. The Hopf-Rinow Theorem.- 3. Spaces with curvature bounded from above.- 4. The Hadamard-Cartan Theorem.- 5. Hadamard spaces.- II.… Expand

Topologies on Closed and Closed Convex Sets

- Mathematics
- 1993

Preface. 1. Preliminaries. 2. Weak Topologies determined by Distance Functionals. 3. The Attouch--Wets and Hausdorff Metric Topologies. 4. Gap and Excess Functionals and Weak Topologies. 5. The Fell… Expand

Geometry of the Space of Phylogenetic Trees

- Mathematics, Computer Science
- Adv. Appl. Math.
- 2001

We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring… Expand

Infinite Dimensional Analysis: A Hitchhiker’s Guide

- Mathematics
- 1994

This text was born out of an advanced mathematical economics seminar at Caltech in 1989-90. We realized that the typical graduate student in mathematical economics has to be familiar with a vast… Expand

Metric Structures for Riemannian and Non-Riemannian Spaces

- Mathematics
- 1999

Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.-… Expand

Some properties of the Hausdorff distance in metric spaces

- Mathematics
- 1990

Some properties of the Hausdorff distance in complete metric spaces are discussed. Results obtained in this paper explain ideas used in the theory of measures of noncompactness

Nonexpansive Retracts in Banach Spaces

- Mathematics
- 2007

We study various aspects of nonexpansive retracts and retractions in certain Banach and metric spaces, with special emphasis on the compact nonexpansive envelope property. 2000 Mathematics Subject… Expand

On hyperbolic groups

- Mathematics
- 2006

Abstract We prove that a δ-hyperbolic group for δ < ½ is a free product F * G 1 * … * Gn where F is a free group of finite rank and each Gi is a finite group.