Riemannian barycentres and geodesic convexity
UNSPECIFIED (1999) Riemannian barycentres and geodesic convexity. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 127 (Part 2). pp. 253-269. ISSN 0305-0041Full text not available from this repository.
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|
|Number of Pages:||17|
|Page Range:||pp. 253-269|
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