Geometric Measure Theory

By T. O’Neil. 2002

3,943 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/45ae22d4c317d8d8623f76bb3f0d3bace4913414

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Citations in the corpus

Nearest-Neighbor Searching and Metric Space Dimensions (2005, background): “Such a space is also called a growth-restricted metric[KR02] or Federer measure or a diametrically regular measure [Fed69].”
50 years sets with positive reach -- a survey. (2008, unknown): “[5] H.”
Level crossings and other level functionals of stationary Gaussian processes (2006, background): “) (u) be the length of {(s,t) ∈ Q(T) : X s,t= u} the level curve at level ufor the random ﬁeld X, Q(T) being the square [−T,T]×[−T,T] and |Q(T)| its Lebesgue measure. By theorem 3.2.5 of Federer (see [53], p.244), we have for g∈ C(IR) Z∞ −∞ g(u)LX Q(T) (u)du= ZZ Q(T) g(X s,t)||∇X s,t||dsdt. We assum”
The singular set of mean curvature flow with generic singularities (2014, unknown): “4 See, for instance [7,19,25,36,37,40,45].”
A local Steiner–type formula for general closed sets and applications (2004, methodology): “If it were, then the boundary of each compact set with vanishing volume would admit a finite (d − 1)-dimensional Minkowski content (see [5], [3]).”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “The reader can look at [162] for more details.”
Gauge theory and calibrated geometry, I (2000, methodology): “e work of D. Priess [P], since Θ(ν,·) exists almost everywhere and νis Borel regular. However, for the reader’s convenience, we give a direct proof here by using the structure theorem of Federer (cf. [Fe], [Li]). We may write S = Su ∪ Sr, where Sr is a rectiﬁable set and Su is a purely unrectiﬁable ”
Functions of bounded variation on “good” metric spaces (2003, methodology): “The paper is structured as follows: in Section 2 we will give the main definitions present our setting; in Section 3 we give the definition ofBV functions and give the main properties; in Section 4 we define the sets of finite perimeter in the same w Caccioppoli [7] and De Giorgi [10] did (see also ”
Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, methodology): “Federer’s coarea formula gives, see [39], ∫”
Regularity for solutions of the total variation denoising problem (2011, background): “The first equality just follows from Besicovitch’s derivation theorem for Radon measures, while the second from a careful study of the reduced boundary, see [16] and Section 3 in [3].”
Degeneration of Riemannian metrics under Ricci curvature bounds (2013, methodology): “For the facts concerning Hausdorff measure which are recalled below, we refer to Chapter 11 of [Gi], or Chapter 1 of [Sim]; see also [Fe], [Ma].”
Pull-back of currents by holomorphic maps (2006, unknown): “ Bk × Bk′. Then τ− τ0 is a positive current with support in Σ. Lemma 3.1 applied to ddcT n implies that ddc(τ−τ 0) has order zero. The support theorem in [1] extends Federer’s theorem on ﬂat currents [16] to currents of order zero with ddc of order zero. It implies that τ− τ0 is a current of Σ. Let ”
Finite volume flows and Morse theory (2001, background): “In particular, d[Sp] has finite mass, and so by [F; 4.2.16(2)] it is a rectifiable current.”
A multivariate Berry–Esseen theorem with explicit constants (2018, methodology): “Proposition 3.2 (Federer [14], Corollary 3.2.32).”