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Dilation volumes of sets of finite perimeter

  • Markus Kiderlen (a1) and Jan Rataj (a2)

Abstract

In this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilation AtQ as t converges to 0. Here A and Q are subsets of n-dimensional Euclidean space, A has finite perimeter, and Q is finite. If Q consists of two points only, n and n+u, say, this derivative coincides up to a sign with the directional derivative of the covariogram of A in direction u. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of A. We extend this result to finite sets Q and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at 0. The proofs are based on an approximation of the indicator function of A by smooth functions of bounded variation.

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Corresponding author

* Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, DK-8000 Aarhus C, Denmark. Email address: kiderlen@math.au.dk
** Postal address: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic. Email address: rataj@karlin.mff.cuni.cz

References

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[1]Amar, M. and Bellettini, G. (1994).A notion of total variation depending on a metric with discontinuous coefficients.Ann. Inst. H. Poincaré Anal. Non Linéaire 11,91133.
[2]Ambrosio, L.,Fusco, N. and Pallara, D. (2000).Functions of Bounded Variation and Free Discontinuity Problems.Oxford University Press,New York.
[3]Caccioppoli, R. (1952).Misura e integrazione sugli insiemi dimensionalmente orientati.Rend. Accad. Naz. Lincei Ser. 8 12,311 (in Italian).
[4]Caccioppoli, R. (1952).Misura e integrazione sugli insiemi dimensionalmente orientati. II.Rend. Accad. Naz. Lincei Ser. 8 12,137146 (in Italian).
[5]Chambolle, A.,Lisini, S. and Lussardi, L. (2014).A remark on the anisotropic outer Minkowski content.Adv. Calc. Var. 7,241266.
[6]De Giorgi, E. (1954).Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni.Ann. Mat. Pura Appl. 36,191213 (in Italian).
[7]De Giorgi, E. (1955).Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio ad r dimensioni.Ricerche Mat. 4,95113 (in Italian).
[8]De Giorgi, E. (1958).Sulla proprietá isoperimentrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita.Mem. Accad. Naz. Lincei Mem. Ser. 8 5,3344 (in Italian).
[9]Dellacherie, C. and Meyer, P.-A. (1978).Probabilities and Potential (North-Holland Math. Stud. 29).North-Holland,Amsterdam.
[10]Federer, H. (1969).Geometric Measure Theory.Springer,New York.
[11]Galerne, B. (2011).Computation of the perimeter of measurable sets via their covariogram. Applications to random sets.Image Anal. Stereol. 30,3951.
[12]Galerne, B. and Lachièze-Rey, R. (2015).Random measurable sets and covariogram realizability problems.Adv. Appl. Prob. 47,611639.
[13]Hug, D. and Last, G. (2000).On support measures in Minkowski spaces and contact distributions in stochastic geometry.Ann. Prob. 28,796850.
[14]Kiderlen, M. and Rataj, J. (2006).On infinitesimal increase of volumes of morphological transforms.Mathematika 53,103127.
[15]Lussardi, L. and Villa, E. (2016).A general formula for the anisotropic outer Minkowski content of a set.Proc. R. Soc. Edinburgh A 146,393413.
[16]Matheron, G. (1965).Les Variables Régionalisées et Leur Estimation.Masson,Paris.
[17]Pollard, D. (1984).Convergence of Stochastic Processes.Springer,New York.
[18]Rataj, J. (2004).On set covariance and three-point test sets.Czechoslovak Math. J. 54,205214.
[19]Rataj, J. (2015).Random sets of finite perimeter.Math. Nachr. 288,10471056.
[20]Schneider, R. (1974).Additive transformationen konvexer Körper.Geometriae Dedicata 3,221228.
[21]Schneider, R. (2014).Convex Bodies: the Brunn-Minkowski Theory (Encyclopedia Math. Appl. 151),2nd edn.Cambridge University Press.
[22]Schneider, R. and Weil, W. (2008).Stochastic and Integral Geometry.Springer,Berlin.
[23]Villa, E. (2010).Mean densities and spherical contact distribution function of inhomogeneous Boolean models.Stoch. Anal. Appl. 28,480504.
[24]Ziemer, W. P. (1989).Weakly Differentiable Functions.Springer,New York.

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Dilation volumes of sets of finite perimeter

  • Markus Kiderlen (a1) and Jan Rataj (a2)

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