Normal embeddings of semialgebraic sets.

Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces

“…Only after the introduction of Lregular decompositions by Kurdyka ([12]) and Parusiński ([17]), there has been some progress in understanding the metric structure of subanalytic sets (e.g. [10], [4], [13], [16], [8], [18], [1], [2]). However, it seems that many natural questions are still out of reach.…”

Metrically Un-knotted Corank 1 Singularities of Surfaces in $$\mathbb {R}^4$$ R 4

“…By the results of Kurdyka and Orro or Birbrair and Mostowski (see [2] and [3]) there exists a semialgebraic metric d P bi-Lipschitz equivalent to the intrinsic metric d inner . That is why we can also define the (intrinsic) order of contact of γ 1 and γ 2 as the order of the function d inner (γ 1 (t), γ 2 (t)), which is denoted by tord inn (γ 1 , γ 2 ).…”

Thin-thick decomposition for real definable isolated singularities

“…We recall that a subset of R n is normally embedded if the outer metric and the inner metric are bi-Lipschitz equivalent [6]. Reading the proof of this theorem, we observe that this result can be achieved definably if we start with definable data.…”

“…According to [6], there exist a subset X ′ ⊂ R m and a map Φ : X ′ → X such that (1) X ′ is normally embeddd in R m ; The subset X ′ is called a normal embedding of X . (2) The map Φ is definable and bi-Lipschitz with respect to the inner metric.…”