Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces

Abstract

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, blowups 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.