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Abstract: We study singular semialgebraic, subanalytic and complex analytic
sets from a metric viewpont. Each connected set of these types
can be considered as a metric space with different natural metrics.
The most interesting metrics are induced (euclidian) metric and
intrinsic metric. A set is called normally embedded if these two
metrics are bi-Lipschitz equivalent. The main result of the talk
is the following :
Every closed semialgebraic set is intrinsically bi-Lipschitz
equivalent to some normally embedded semialgebraic set.
This result does not admit a complex analog. We present an example
of a complex analytic set which is not intrinsically bi-Lipschitz
equivalent to any normally embedded complex analytic set.
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