Set estimation: another bridge between statistics and geometry

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Statistical Analysis and Parameter Selection for Mapper (2017, background): “All of them can be found, together with their proofs, in Dey and Wang (2013) and Carrière and Oudot (2015).”
Topological Data Analysis (2016, methodology): “The red lines show an estmate of S using the methods in Section 3.3. d-dimensional but it does converge to S in the following sense (Cuevas, 2009; Fasy et al., 2014b; Niyogi et al., 2008; Cuevas et al., 2001; Chazal et al., 2014b).”
Plugin estimation of smooth optimal transport maps (2021, background): “…exist 0, δ0 > 0 such that for all x ∈ Ω and ∈ (0, 0), L(B(x, ) ∩ Ω) ≥ δ0L(B(x, )), where recall that L denotes the Lebesgue measure on Rd. Condition (S2) arises frequently in the literature on statistical set estimation (Cuevas and Fraiman, 1997; Cuevas, 2009), and prevents Ω from admitting cusps.”