The Infinitisimal Erosions

  • G. Matheron
Conference paper
Part of the Lecture Notes in Biomathematics book series (LNBM, volume 23)


On the space C(R) of the compact convex sets in Rn, the erosion by the homctrctic sets ρk(of a fixed K ∈ C(R) constitutes α semi-group, the generator of which is defined by the relationship \( K(A)=\lim (A\theta \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A}\rho)/\rho \), when ρ ↓ 0 (A = A θ ρ.\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{K}\)). This generator is called “infinitesimal erosion”. K(A) depends only on the support SA of the surface measure associated with A only. More precisely: K(A) is the largest convex on SA. As an application of this theorem, one solves the equation X θ \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{K}\) = A (A, K known, X ∈ C(R) unknown).


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  1. [1]
    G. Matheron (1977): La formule de Steiner pour les érosions (à paraître dans Adv. in Appl. Prob.).Google Scholar
  2. [2]
    I. Minkowski (1903):,Volumen und Oberfl.che. Math. Ann., Vol. 57, PP. 447–495.MathSciNetCrossRefGoogle Scholar
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    R.T. Rockafellar (1972): Convex Analysis. Princeton University Press.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

  • G. Matheron
    • 1
  1. 1.Centre de Morphologie Mathematique E.M.P.FontainebleauFrance

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