Tubular neighborhoods in Euclidean spaces

Fu, Joseph H. G.

doi:10.1215/s0012-7094-85-05254-8

Cited by 113 publications

(105 citation statements)

References 15 publications

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“…with g = 6 , D = log 3 2, and p = 2π/ log 3. It is this form of the tube formula which is closer in appearance to the following examples.…”

Section: 2mentioning

confidence: 99%

“…See [1, §6.6-6.9] and the book [9]. Federer's tube formula has since been extended in various directions by a number of researchers in integral geometry and geometric measure theory, including [36], [37], [42], [43], [6], [7], [38], and most recently (and most generally) in [13]. The books [9] and [37] contain extensive endnotes with further information and many other references.…”

Section: Introductionmentioning

confidence: 99%

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Tube Formulas and Complex Dimensions of Self-Similar Tilings

Lapidus

Pearse

2010

Acta Appl Math

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Abstract. We use the self-similar tilings constructed in [32] to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubular zeta function and hence develop a tube formula for self-similar tilings in R d . The resulting power series in ε is a fractal extension of Steiner's classical tube formula for convex bodies K ⊆ R d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i = 0, 1, . . . , d − 1, just as Steiner's does. However, our formula also contains a term for each complex dimension. This provides further justification for the term "complex dimension". It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in [30].

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“…with g = 6 , D = log 3 2, and p = 2π/ log 3. It is this form of the tube formula which is closer in appearance to the following examples.…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tube Formulas and Complex Dimensions of Self-Similar Tilings

Lapidus

Pearse

2010

Acta Appl Math

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“…Remark 1.1. Twenty years ago, Fu proved in [6] that the set of all critical values of the distance function from a compact subset in R 3 is of Lebesgue measure zero. Following Whitney [17], S. Ferry [5] gave a counterexample to this result in dimension greater than three.…”

Section: Introductionmentioning

confidence: 99%

A Morse-Sard theorem for the distance function on Riemannian manifolds

Rifford

2004

manuscripta mathematica

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“…In [3], Ferry showed, among other results, that ∂ (A) is a 1-manifold for almost all . Fu [4] generalized Ferry's results, and proved that for all outside a compact set of zero 1/2-dimensional Hausdorff measure, ∂ (A) is a Lipschitz 1-manifold. Papers [3] and [4] include theorems in higher dimensional Euclidean spaces; the work for dimensions n ≥ 3 is more demanding.…”

Section: Introductionmentioning

confidence: 69%

Sets of constant distance from a Jordan curve

Vellis¹,

Wu²

2014

Ann. Acad. Sci. Fenn. Math.

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Abstract. We study the -level sets of the signed distance function to a planar Jordan curve Γ, and ask what properties of Γ ensure that the -level sets are Jordan curves, or uniform quasicircles, or uniform chord-arc curves for all sufficiently small . Sufficient conditions are given in term of a scaled invariant parameter for measuring the local deviation of subarcs from their chords. The chordal conditions given are sharp.

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Tubular neighborhoods in Euclidean spaces

Cited by 113 publications

References 15 publications

Tube Formulas and Complex Dimensions of Self-Similar Tilings

Tube Formulas and Complex Dimensions of Self-Similar Tilings

A Morse-Sard theorem for the distance function on Riemannian manifolds

Sets of constant distance from a Jordan curve

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