Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T03:58:52.137Z Has data issue: false hasContentIssue false
  • English
  • Français

Dilation volumes of sets of finite perimeter

Part of: Functions of several variables Geometric probability and stochastic geometry Classical measure theory

Published online by Cambridge University Press:  29 November 2018

Markus Kiderlen  and
Jan Rataj
Markus Kiderlen*
Affiliation:
University of Aarhus
Jan Rataj*
Affiliation:
Charles University, Prague
*
* Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, DK-8000 Aarhus C, Denmark. Email address: [email protected]
** Postal address: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilation A⊕tQ 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.

Keywords

Bounded variationcontact distribution functiondilation volumedirectional variationsets of finite perimeterstationary random setsurface area measure

MSC classification

Primary: 26B30: Absolutely continuous functions, functions of bounded variation
Secondary: 28A75: Length, area, volume, other geometric measure theory 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1095 - 1118
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[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 11,91–133.Google Scholar
[2]Ambrosio, L.,Fusco, N. and Pallara, D. (2000).Functions of Bounded Variation and Free Discontinuity Problems.Oxford University Press,New York.Google Scholar
[3]Caccioppoli, R. (1952).Misura e integrazione sugli insiemi dimensionalmente orientati.Rend. Accad. Naz. Lincei Ser. 8 12,3–11 (in Italian).Google Scholar
[4]Caccioppoli, R. (1952).Misura e integrazione sugli insiemi dimensionalmente orientati. II.Rend. Accad. Naz. Lincei Ser. 8 12,137–146 (in Italian).Google Scholar
[5]Chambolle, A.,Lisini, S. and Lussardi, L. (2014).A remark on the anisotropic outer Minkowski content.Adv. Calc. Var. 7,241–266.Google Scholar
[6]De Giorgi, E. (1954).Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni.Ann. Mat. Pura Appl. 36,191–213 (in Italian).Google Scholar
[7]De Giorgi, E. (1955).Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio ad r dimensioni.Ricerche Mat. 4,95–113 (in Italian).Google Scholar
[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 5,33–44 (in Italian).Google Scholar
[9]Dellacherie, C. and Meyer, P.-A. (1978).Probabilities and Potential (North-Holland Math. Stud. 29).North-Holland,Amsterdam.Google Scholar
[10]Federer, H. (1969).Geometric Measure Theory.Springer,New York.Google Scholar
[11]Galerne, B. (2011).Computation of the perimeter of measurable sets via their covariogram. Applications to random sets.Image Anal. Stereol. 30,39–51.Google Scholar
[12]Galerne, B. and Lachièze-Rey, R. (2015).Random measurable sets and covariogram realizability problems.Adv. Appl. Prob. 47,611–639.Google Scholar
[13]Hug, D. and Last, G. (2000).On support measures in Minkowski spaces and contact distributions in stochastic geometry.Ann. Prob. 28,796–850.Google Scholar
[14]Kiderlen, M. and Rataj, J. (2006).On infinitesimal increase of volumes of morphological transforms.Mathematika 53,103–127.Google Scholar
[15]Lussardi, L. and Villa, E. (2016).A general formula for the anisotropic outer Minkowski content of a set.Proc. R. Soc. Edinburgh A 146,393–413.Google Scholar
[16]Matheron, G. (1965).Les Variables Régionalisées et Leur Estimation.Masson,Paris.Google Scholar
[17]Pollard, D. (1984).Convergence of Stochastic Processes.Springer,New York.Google Scholar
[18]Rataj, J. (2004).On set covariance and three-point test sets.Czechoslovak Math. J. 54,205–214.Google Scholar
[19]Rataj, J. (2015).Random sets of finite perimeter.Math. Nachr. 288,1047–1056.Google Scholar
[20]Schneider, R. (1974).Additive transformationen konvexer Körper.Geometriae Dedicata 3,221–228.Google Scholar
[21]Schneider, R. (2014).Convex Bodies: the Brunn-Minkowski Theory (Encyclopedia Math. Appl. 151),2nd edn.Cambridge University Press.Google Scholar
[22]Schneider, R. and Weil, W. (2008).Stochastic and Integral Geometry.Springer,Berlin.Google Scholar
[23]Villa, E. (2010).Mean densities and spherical contact distribution function of inhomogeneous Boolean models.Stoch. Anal. Appl. 28,480–504.Google Scholar
[24]Ziemer, W. P. (1989).Weakly Differentiable Functions.Springer,New York.Google Scholar