UNSPECIFIED (1999) Riemannian barycentres and geodesic convexity. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 127 (Part 2). pp. 253-269. ISSN 0305-0041

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Abstract

Consider a closed subset of a complete Riemannian manifold, such that all geodesics with end-points in the subset are contained in the subset and the subset has boundary of codimension one. Is it the case that Riemannian barycentres of probability measures supported by the subset must also lie in the subset? It is shown that this is the case for 2-manifolds but not the ease in higher dimensions: a counterexample is constructed which is a conformally-Euclidean 3-manifold, for which geodesics never self-intersect and indeed cannot turn by too much (so small geodesic balls satisfy a geodesic convexity condition), but is such that a probability measure concentrated on a single point has a barycentre at another point.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Journal or Publication Title:	MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY
Publisher:	CAMBRIDGE UNIV PRESS
ISSN:	0305-0041
Date:	September 1999
Volume:	127
Number:	Part 2
Number of Pages:	17
Page Range:	pp. 253-269
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/14115

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