Outer Minkowski content for some classes of closed sets

Ambrosio, Luigi; Colesanti, Andrea; Villa, Elena

doi:10.1007/s00208-008-0254-z

Cited by 97 publications

(143 citation statements)

References 15 publications

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“…1 thus implies that if A is a full-dimensional compact U PR -set then V u (A) = 2T P u (A). This identity is the directional equivalent of a recent result due to Ambrosio, Colesanti, and Villa (2008): a fulldimensional compact set with positive reach A satisfies Per(A) = 2Φ d−1 (A) (Ambrosio et al, 2008, Theorem 9) (one could directly prove that V u (A) = 2T P u (A) by using the techniques developed in (Ambrosio et al, 2008) and (Rataj, 2004)). Since Eq.…”

Section: Introductionmentioning

confidence: 74%

Computation of the Perimeter of Measurable Sets via Their Covariogram. Applications to Random Sets

Galerne¹

2011

Image Anal Stereol

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The covariogram of a measurable set A ⊂ R d is the function g A which to each y ∈ R d associates the Lebesgue measure of A ∩ (y + A). This paper proves two formulas. The first equates the directional derivatives at the origin of g A to the directional variations of A. The second equates the average directional derivative at the origin of g A to the perimeter of A. These formulas, previously known with restrictions, are proved for any measurable set. As a by-product, it is proved that the covariogram of a set A is Lipschitz if and only if A has finite perimeter, the Lipschitz constant being half the maximal directional variation. The two formulas have counterparts for mean covariogram of random sets. They also permit to compute the expected perimeter per unit volume of any stationary random closed set. As an illustration, the expected perimeter per unit volume of stationary Boolean models having any grain distribution is computed.

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Section: Introductionmentioning

confidence: 74%

Computation of the Perimeter of Measurable Sets via Their Covariogram. Applications to Random Sets

Galerne¹

2011

Image Anal Stereol

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“…As a matter of fact, it can be seen as the stochastic version of Eq. 10, which, in many applications, is satisfied with η(·) = H n ( A ∩ · ) for some closed set A ⊇ A, as proved in (Ambrosio et al, 2000, p. 111) (see also Ambrosio et al, 2008).…”

Section: Remarkmentioning

confidence: 74%

“…14 is not satisfied, and λ ∂ Ξ = σ Ξ for the corresponding Boolean model Ξ. Furthermore, it can be shown (Ambrosio et al, 2008) that the class of subsets A of R d such that (A 1 ∪A 2 )). Noticing that, by condition Eq.…”

Section: Proposition 11 Let ξ Be a Boolean Model As In Thementioning

confidence: 99%

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On the Specific Area of Inhomogeneous Boolean Models. Existence Results and Applications

Villa¹

2011

Image Anal Stereol

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The problem of the evaluation of the so-called specific area of a random closed set, in connection with its mean boundary measure, is mentioned in the classical book by Matheron on random closed sets (Matheron, 1975, p. 50); it is still an open problem, in general. We offer here an overview of some recent results concerning the existence of the specific area of inhomogeneous Boolean models, unifying results from geometric measure theory and from stochastic geometry. A discussion of possible applications to image analysis concerning the estimation of the mean surface density of random closed sets, and, in particular, to material science concerning birth-and-growth processes, is also provided.

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“…If A is convex or with sufficiently regular boundary then the equality (4) also holds for t = 0. For precise statements and comparisons between the outer Minkowski surface area and other measurements of ∂(A + tB n 2 ), like the Hausdorff measure, see [2].…”

Section: Regularity Properties Of the Parallel Volumementioning

confidence: 99%

On the analogue of the concavity of entropy power in the Brunn–Minkowski theory

Fradelizi

Marsiglietti

2014

Advances in Applied Mathematics

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Elaborating on the similarity between the entropy power inequality and the Brunn-Minkowski inequality, Costa and Cover conjectured in On the similarity of the entropy power inequality and the BrunnMinkowski inequality (IEEE Trans. Inform. Theory 30 (1984), no. 6, 837-839) the 1 n -concavity of the outer parallel volume of measurable sets as an analogue of the concavity of entropy power. We investigate this conjecture and study its relationship with geometric inequalities.

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Outer Minkowski content for some classes of closed sets

Cited by 97 publications

References 15 publications

Computation of the Perimeter of Measurable Sets via Their Covariogram. Applications to Random Sets

Computation of the Perimeter of Measurable Sets via Their Covariogram. Applications to Random Sets

On the Specific Area of Inhomogeneous Boolean Models. Existence Results and Applications

On the analogue of the concavity of entropy power in the Brunn–Minkowski theory

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