Integral and current representation of Federer's curvature measures

Zähle, M.

doi:10.1007/bf01195026

Cited by 115 publications

(136 citation statements)

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“…Of course, a closed convex set A satisfies reach(A) = ∞. An extension of the local Steiner formula (1.3) to sets with positive reach was established by Federer [4], a simpler and more general approach was later developed by M. Zähle [19]. To state such a local Steiner formula, let A ⊂ R d satisfy reach(A) ≥ r. Then, for any measurable bounded function f : R d × S d−1 → R with compact support and t ∈ (0, r),…”

Section: Theorem 2 Let a ⊂ R D Be Non-empty And Closed Assume That mentioning

confidence: 99%

Does polynomial parallel volume imply convexity?

Heveling

Hug

2004

Mathematische Annalen

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Abstract. For a non-empty compact set A ⊂ R d , d ≥ 2, and r ≥ 0, let A ⊕r denote the set of points whose distance from A is r at the most. It is well-known that the volume, V d (A ⊕r ), of A ⊕r is a polynomial of degree d in the parameter r if A is convex. We pursue the reverse question and ask whether A is necessarily convex if V d (A ⊕r ) is a polynomial in r. An affirmative answer is given in dimension d = 2, counterexamples are provided for d ≥ 3. A positive resolution of the question in all dimensions is obtained if the assumption of a polynomial parallel volume is strengthened to the validity of a (polynomial) local Steiner formula. (2000): 52A38, 28A75, 52A22, 53C65 Mathematics Subject Classification

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Section: Theorem 2 Let a ⊂ R D Be Non-empty And Closed Assume That mentioning

confidence: 99%

Does polynomial parallel volume imply convexity?

Heveling

Hug

2004

Mathematische Annalen

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“…If µ is an invariant valuation of degree k < 2n, then we represent µ by ω ∈ Ω 2n−1 (SC n ) SU (n) , i.e. if nc(K) is the normal cycle of K (compare [28] for the normal cycle of compact convex sets), then…”

Section: Weight Decompositionmentioning

confidence: 99%

A Hadwiger-Type Theorem for the Special Unitary Group

Bernig

2009

Geom. Funct. Anal.

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“…For sets with positive reach, curvature measures have an integral representation very similar to the one mentioned above for smooth manifolds where the principal curvatures are replaced by the so-called generalized principal curvatures, which are given as limits of corresponding principal curvatures in the parallel sets K ε (cf. [35]). …”

Section: Introductionmentioning

confidence: 99%