Convexity in finite metric spaces
type: Item
instance of: scholarly article
title: Convexity in finite metric spaces
zbMATH Open document ID: 0809.52002
DOI: 10.1007/BF01278473
published in: Geometriae Dedicata
publication date: 1994-10-30T00:00:00Z
review text: A subset \(S\) of a metric space \((X,d)\) is called \(d\)-convex if for every pair of points \(x, y \in S\) all the points between \(x\) and \(y\) belong to \(S\). Let \({\mathfrak C}_ d\) be the family of all \(d\)-convex subsets of \((X,d)\). Then \((X, {\mathfrak C}_ d)\) is a convexity structure (Prop. 1), i.e., \(\emptyset \in {\mathfrak C}_ d\) and \({\mathfrak C}_ d\) is closed under intersection of any subfamily. The paper concerns two kinds of embeddings: isometric and convex (i.e. convexity preserving) embeddings. In particular, it concerns embeddability of (finite) metric spaces into \(\mathbb{R}^ n\) with either Euclidean metric \(d_ e\) or ``max'' metric \(d_ m\), or into the so called Hamming space \((\{0, 1\}^ n,h)\) with \(h(x,y) = \sum | x_ i - y_ i |\). The paper contains many interesting open problems and a number of results.
reviewed by: Maria Moszyńska
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 51K99
Mathematics Subject Classification ID: 54C25
zbMATH DE Number: 679544
zbMATH Keywords: \(d\)-convex set
zbMATH Keywords: finite space
zbMATH Keywords: isometric embedding
zbMATH Keywords: convex embedding
zbMATH Keywords: metric space
zbMATH Keywords: convexity structure
author: Eike Hertel
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01278473
OpenAlex ID: W2075792703
description: scientific article; zbMATH DE number 679544