Sunday, 15 January 2012

mg.metric geometry - When completion of locally compact length space is locally compact?

A necessary and sufficient condition (but I do not feel satisfied with that) for the locally compact length space X to have a locally compact completion is that there exists some r>0 such that each ball of radius r in X is totally bounded.



In fact, if the condition holds closed balls of radius r/2 in overlineX are compact.
On the other hand, suppose that overlineX is locally compact. Then, as it is a complete length space, it is proper (this is called the Hopf-Rinow Theorem in the book by Bridson and Haefliger). This should imply that balls of any radius in X are totally bounded.



The main reason why I am not satisfied with it is that the proof that the condition is sufficient does not use that X is a length space, so this is not really the answer to what you asked. I thought it might be relevant, anyway...

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