On the expected surface area of the Wiener sausage

Abstract. We show that whenever the q-dimensional Minkowski content of a subset A ⊂ R d exists and is finite and positive, then the "S-content" defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in R d , d 3.

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“…Recently it was shown that, as expected, the mean surface area of the Wiener sausage satisfies (see [4], [5])…”

Section: Introductionsupporting

confidence: 64%

Almost sure asymptotic behaviour of the r-neighbourhood surface area of Brownian paths

Honzl

Rataj

2012

Czech Math J

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“…Proof. Rataj et al in [18], theorem 3.3, showed that V ′ A (t) exists for every t ≥ diam(A), thus we have for every t ≥ diam(A)…”

Section: The Preceding Theorem Is Still Valid If One Replaces B Nmentioning

confidence: 75%

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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“…as l → ∞ due to the integrability of γ 2 11 . The following statement gives a sufficient condition for (36).…”

Section: Preliminary Results From Linear Regressionmentioning

confidence: 99%

“…Corollary 3.6. If Q l = σ 2 I l for some σ > 0, where I l is the identity matrix, then condition (36) in Theorem 3.5 is satisfied for all t = 0 provided that…”

Section: Preliminary Results From Linear Regressionmentioning

confidence: 99%

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Estimation of fractal dimension and fractal curvatures from digital images

Spodarev

Straka

Winter

2015

Chaos, Solitons & Fractals

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Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e. volume, surface area, Euler characteristic etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behaviour, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.

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On the expected surface area of the Wiener sausage

Cited by 19 publications

References 23 publications

Almost sure asymptotic behaviour of the r-neighbourhood surface area of Brownian paths

Almost sure asymptotic behaviour of the r-neighbourhood surface area of Brownian paths

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

Estimation of fractal dimension and fractal curvatures from digital images

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