Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces

Abstract

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.

Thin-thick decomposition for real definable isolated singularities

“…We defineΦ(v) as the unit tangent vector of Φ(γ) at 0. This construction is analogous to the construction presented by Bernig and Lytchak [1]. It is also clear thatΦ is a definable map.…”

“…Indeed, this map is bi-Lipschitz on each pancake. The results of [1] and [5] imply thatΦ is bi-Lipschitz on each pancake. Since, a pancake decomposition is finite, the mapΦ is also finite.…”

“…The topology of the components of the thick zone are related to the topology of the corresponding parts of the metric tangent cone (see [1] or [5]). Finally we show that the topology of the thin-thick decomposition of the set, equipped with the multiplicity, is an invariant for the blow-spherical equivalence.…”

Lipschitz regular complex algebraic sets are smooth

“…Let us suppose that h(x 0 ) = 0. Then the derivative of h, dh : |C x 0 X| → T 0 B, defined by Bernig and Lytchak [1] (see also [2]), is a bi-Lipschitz homeomorphism between the reduced tangent cones |C x 0 X| and T 0 B = R N . In particular, it proves that |C x 0 X| is also bi-Lipschitz regular at x 0 .…”

Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

“…This was proved by Bernig and Lytchak [1] under the additional assumption that the bi-Lipschitz homeomorphism is also subanalytic (see also the paper of Birbrair, Fernandes and Neumann [4] for an extremely simple proof). This result gives a positive answer to Zariski's question for bi-Lipschitz homeomorphism.…”