Real Algebraic and Analytic Geometry
Submission: 2004, February 25.
It is shown that the Gromov-Hausdorff limit of a subanalytic $1$-parameter family of compact connected sets (endowed with the inner metric) exists. If the family is semialgebraic, then the limit space can be identified with a semialgebraic set over some real closed field. Different notions of tangent cones (pointed Gromov-Hausdorff limits, blow-ups and Alexandrov cones) for a closed connected subanalytic set are studied and shown to be naturally isometric. It is shown that geodesics have well-defined Euclidean directions at each point.
Mathematics Subject Classification (2000): 53C22, 32B20.
Keywords and Phrases: Tangent spaces, Gromov-Hausdorff convergence, subanalytic sets, geodesics.
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