On a link criterion for Lipschitz normal embeddings among definable sets. (English) Zbl 1532.14022
On some connected definable set \(X\subset \mathbb{R}^{n}\) one has two natural metrics: the restriction of the Euclidean metric of the ambient space called outer metric, and the inner metric, where the distance between two points in \(X\) is the infimum of the lengths of rectifiable curves within \(X\) connecting these points. One says that \(X\) is “Lipschitz normally embedded” if these two metrics are equivalent.
It was shown in [R. Mendes and J. E. Sampaio, “On the link of Lipschitz normally embedded sets”, Preprint, arXiv:2101.05572] that a subanalytic space germ \((X,0)\) is Lipschitz normally embedded if and only if its link is Lipschitz normally embedded. The author shows that this equivalence still holds for definable germs in any o-minimal structure on \((\mathbb{R}, +, \cdot)\). He also produces an interesting example which gives a negative answer to a question concerning the recently introduced “moderately discontinuous homology”.
It was shown in [R. Mendes and J. E. Sampaio, “On the link of Lipschitz normally embedded sets”, Preprint, arXiv:2101.05572] that a subanalytic space germ \((X,0)\) is Lipschitz normally embedded if and only if its link is Lipschitz normally embedded. The author shows that this equivalence still holds for definable germs in any o-minimal structure on \((\mathbb{R}, +, \cdot)\). He also produces an interesting example which gives a negative answer to a question concerning the recently introduced “moderately discontinuous homology”.
Reviewer: Mihai-Marius Tibar (Lille)
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