A geometric second-order-rectifiable stratification for closed subsets of Euclidean space

Menne, Ulrich; Santilli, Mario

doi:10.2422/2036-2145.201703_021

Cited by 12 publications

(19 citation statements)

References 10 publications

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“…In fact, we can choose v = a − x for a ∈ A such that |x − a| = r. Therefore (2) comes from [MS17,4.12]. Notice that [MS17, 4.12] also implies that S(A, r) is countably n − 1 rectifiable, a piece of information already contained in (1).…”

Section: Level Sets Of Distance Functionmentioning

confidence: 99%

“…A is a closed subset of R n . For each 0 ≤ m ≤ n, we define the m-th stratum of A by 3 It is proved in [MS17,4.12] that A (m) is a countably m rectifiable Borel set which can be H m almost covered by the union of a countable family of m dimensional submanifolds of R n of class 2. Finally one may use Coarea formula to infer that (N (A, a)) < ∞} for m = 0, .…”

Section: Definition Supposementioning

confidence: 99%

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Fine properties of the curvature of arbitrary closed sets

Santilli

2019

Annali di Matematica

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Given an arbitrary closed set A of R n , we establish the relation between the eigenvalues of the approximate differential of the spherical image map of A and the principal curvatures of A introduced by Hug-Last-Weil, thus extending a well known relation for sets of positive reach by Federer and Zähle. Then we provide for every m = 1, . . . , n − 1 an integral representation for the support measure µm of A with respect to the m dimensional Hausdorff measure.Moreover a notion of second fundamental form QA for an arbitrary closed set A is introduced so that the finite principal curvatures of A correspond to the eigenvalues of QA. We prove that the approximate differential of order 2, introduced in a previous work of the author, equals in a certain sense the absolutely continuous part of QA, thus providing a natural generalization to higher order differentiability of the classical result of Calderon and Zygmund on the approximate differentiability of functions of bounded variation.MSC-classes 2010. 52A22, 53C65 (Primary); 28A75, 60D05 (Secondary).

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Section: Level Sets Of Distance Functionmentioning

confidence: 99%

Section: Definition Supposementioning

confidence: 99%

Fine properties of the curvature of arbitrary closed sets

Santilli

2019

Annali di Matematica

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“…Proof. If K ⊆ Ω is compact then S ∩ K is (H m , m) rectifiable of class 2 by [MS19,4.12]. Henceforth the conclusion follows from 3.8(1) and 2.6. for H m a.e.…”

Section: Theoremmentioning

confidence: 90%

“…η ∈ N (Γ, a), (iii) trace Q Γ (a, η) ≤ m for H n−m−1 a.e. η ∈ N (Γ, a).Since Γ (m) is (H m , m) rectifiable of class 2 by[MS19, 4.12], it follows from [San19b, 3.23] that condition (i) is satisfied H m almost everywhere on Γ (m) . Moreover, noting 3.8 and 3.10, we infer that(10) ap b Γ (m) (a) • η = −Q Γ (a, η) and trace Q Γ (a, η) ≤ mfor H n−1 a.e.…”

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confidence: 99%

Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

Santilli

2020

Bull. Math. Sci.

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In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, which allows to extend the classical Coarea formula for the Gauss map of smooth varieties, and to introduce for all integral varifolds of bounded mean curvature a natural definition of second fundamental form, whose trace equals the generalized varifold mean curvature. In the second part, we use this machinery to extend a sharp geometric inequality of Almgren to all compact varieties of bounded mean curvature in the viscosity sense and we characterize the equality case. As a consequence we formulate sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled.MSC-classes 2010. 49Q20, 49Q10, 53A07, 53C24, 35D40.Keywords. Bounded mean curvature, varifolds, generalized second fundamental form, generalized Gauss map, Almgren sphere theorem, area blow-up set.1.2 Definition. Suppose A ⊆ R n is a closed set, Ω ⊆ R n is an open set and 1 ≤ m < n is an integer. We say that N (A) satisfies the m dimensional Lusin (N) condition in Ω if and only if the following property holds:is the set of points where A can be touched by a ball from n − m linearly independent directions, (see 2.5).2 In this paper we adopt the terminology in [Alm86, Appendix C] for varifolds; in particular note that the variation function h(V, ·) (i.e. generalized mean curvature of V ) differs from the one adopted in Allard's paper [All72, 4.2] by a sign.

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“…While these papers are a good start, there is still a great deal of opportunity for the use and further development of varifolds. On the theoretical front, there is the work of Menne and collaborators [34,35,36,37,38,28,39,40,41,42]. We want to call special attention to the recent introduction to the idea of a varifold that appeared in the December 2017 AMS Notices [41].…”

Section: Further Explorationmentioning

confidence: 99%

Cubical Covers of Sets in R^n

Paxton¹,

Vixie²

2019

ntmsci

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Wild sets in R n can be tamed through the use of various representations though sometimes this taming removes features considered important. Finding the wildest sets for which it is still true that the representations faithfully inform us about the original set is the focus of this rather playful, expository paper that we hope will stimulate interest in wild sets and their representations, as well as the other two ideas we explore very briefly: Jones' β numbers and varifolds from geometric measure theory.

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A geometric second-order-rectifiable stratification for closed subsets of Euclidean space

Cited by 12 publications

References 10 publications

Fine properties of the curvature of arbitrary closed sets

Fine properties of the curvature of arbitrary closed sets

Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

Cubical Covers of Sets in R^n

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