On the normal cycles of subanalytic sets

Nicolaescu, Liviu I.

doi:10.1007/s10455-010-9241-1

Cited by 6 publications

(13 citation statements)

References 22 publications

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“…Fu [12] gives a formula for the normal cycle in terms of stratified Morse theory; Nicolaescu [24] gives a nice description of the normal cycle from Morse theory.…”

Section: Normal Cyclementioning

confidence: 99%

“…( {σ} such that f −1 (σ) is homeomorphic to U σ ×σ for U σ definable, and, on this inverse image, f acts as projection. 2 For more information, the reader is encouraged to consult [29,10,24].…”

Section: Introductionmentioning

confidence: 99%

“…Perspectives from geometric measure theory and sheaf theory are also relevant to the definition of intrinsic volumes. In this section, we restrict to the o-minimal structure of globally subanalytic sets and use analytic tools based on geometric measure theory, following Alesker [3,4], Fu [12], Nicolaescu [23,24] and many others.…”

Section: Introductionmentioning

confidence: 99%

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Hadwiger’s Theorem for definable functions

Baryshnikov

Wright

2013

Advances in Mathematics

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Hadwiger's Theorem states that En-invariant convex-continuous valuations of definable sets in R n are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable R-valued functions on R n . This generalizes intrinsic volumes to (dual pairs of) non-linear valuations on functions and provides a dual pair of Hadwiger classification theorems. n k=0 c k µ k ,The intrinsic volumes 1 µ k are characterized uniquely by (1) E n invariance, (2) normalization with respect to a closed unit ball, and (3) homogeneity: µ k (λ · A) = λ k (A) for all A ∈ S and λ ∈ R + . These measures generalize Euclidean n-dimensional volume (µ n ) and Euler characteristic (µ 0 ). This paper extends Hadwiger's Theorem to similar valuations on functions instead of sets. Section 2 gives background on the definable (o-minimal) setting that lifts Hadwiger's Theorem to tame, non-convex sets and then to constructible functions; there, we also review the convex-geometric, integral-geometric, and sheaf-theoretic approaches to Hadwiger's Theorem. In Section 4, we consider definable functions Def(R n ) as R-valued functions with tame graphs, and correspondingly define dual pairs of (typically) non-linear "integral" operators · dµ k and · dµ k mapping Def(R n ) → R as generalizations of intrinsic volumes, so that 1 A dµ k = µ k (A) = 1 A dµ k for

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“…Fu [12] gives a formula for the normal cycle in terms of stratified Morse theory; Nicolaescu [24] gives a nice description of the normal cycle from Morse theory.…”

Section: Normal Cyclementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hadwiger’s Theorem for definable functions

Baryshnikov

Wright

2013

Advances in Mathematics

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“…For a precise definition of this object we refer to [1,8,11,12]. Here we will content ourselves with a brief description of its construction.…”

Section: Approximation Of Generic Semi-algebraic Setsmentioning

confidence: 99%

Pixelations of planar semialgebraic sets and shape recognition

Nicolaescu

Rowekamp

2014

Algebr. Geom. Topol.

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We describe an algorithm that associates to each positive real number $r$ and each finite collection $C_r$ of planar pixels of size $r$ a planar piecewise linear set $S_r$ with the following additional property: if $C_r$ is the collection of pixels of size $r$ that touch a given compact semialgebraic set $S$, then the normal cycle of $S_r$ converges to the normal cycle of $S$ in the sense of currents. In particular, in the limit we can recover the homotopy type of $S$ and its geometric invariants such as area, perimeter and curvature measures. At its core, this algorithm is a discretization of stratified Morse theory.Comment: 36 pages, 13 figures, to appear in Algebraic & Geometric Topolog

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“…The aim of this note is to analyze properties of (general) Legendrian cycles, in particular, in connection with its projections to the first and second component. We present a short proof of the uniqueness theorem, due to Fu [4] and proved also in a different setting by Nicolaescu [10], saying that any compactly supported Legendrian cycle is uniquely determined by its restriction to the Gauss curvature form. (Note that this uniqueness property is of particular importance since it makes it possible to introduce Legendrian cycles and curvature measures by approximation with smooth sets, cf.…”

Section: Introductionmentioning

confidence: 99%

Legendrian Cycles and Curvatures

Rataj

Zähle

2014

J Geom Anal

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Properties of general Legendrian cycles T acting in R d × S d−1 are studied. In particular, we give short proofs for certain uniqueness theorems with respect to the projections on the first and second component of such currents: In general, T is determined by its restriction to the Gauss curvature form -this result goes back to J. Fu -and in the full-dimensional case also by the restriction to the surface area form. As a tool a version of the Constancy theorem for Lipschitz submanifolds is shown.2000 Mathematics Subject Classification. 53C65; 28A75; 53A07.

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On the normal cycles of subanalytic sets

Abstract: ABSTRACT. We present a very short complete proof of the existence of the normal cycle of a subanalytic set. The approach is Morse theoretic in flavor and relies heavily on techniques from o-minimal topology.

Cited by 6 publications

References 22 publications

Hadwiger’s Theorem for definable functions

Hadwiger’s Theorem for definable functions

Pixelations of planar semialgebraic sets and shape recognition

Legendrian Cycles and Curvatures

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