Strict $C^1$-triangulations in o-minimal structures

Total Curvature and the Isoperimetric Inequality in Cartan–Hadamard Manifolds

“…These conditions hold, for instance, when Γ is analytic. Indeed in this case cut(Γ) is subanalytic [30,39], which ensures that cut(Γ) admits a C 1 triangulation [50,112]. To prove Lemma Now suppose that equality holds in (1) for some region Ω in a Cartan-Hadamard manifold M .…”

The period isomorphism in the tame geometry

“…Alternatively, we use the same reduction to the affine case as above and then apply [3,Main Theorem] to 𝐺 ⊂ ℝ 𝑁 and the closed subset complementary to the set 𝑈 on which the orientation is defined. This yields a strict 𝐶 1 -triangulation of 𝐺, in particular, all 𝜎 ∶ Δ 𝑑 → 𝐺 are 𝐶 1 as maps to ℝ 𝑁 .…”

“…We choose a definable triangulation of Γ that is globally 𝐶 1 . It exists by Czapła-Pawłucki [3,Main Theorem]. The projection to the first factor is a triangulation (𝐾, Φ) of Δ 𝑑 such that both Φ and 𝜎•Φ are globally 𝐶 1…”

“…Let 𝑈 ⊂ 𝐺 be a definable oriented subset representing the pseudo-orientation. By [2, Proposition 7.6] (or directly [3] for 𝑋 = ℝ 𝑁 ), there is a definable triangulation (𝐾, Φ) of 𝑋 compatible with Ū and Ū − 𝑈 which is globally of class 𝐶 1 . Let 𝜏 1 , … , 𝜏 𝑛 be the 𝑑-dimensional simplices contained in Ū.…”

C1-triangulations of semialgebraic sets

“…The authors would like to thank Tatsuo Suwa for guiding them to this problem and for useful discussions. During preparing the final version of the present paper, we were informed about two recent preprints and ; we also thank G. Comte and M. Hanamura for letting us know of those papers, respectively.…”