Abstract
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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Bibliographie
G. Matheron (1977): La formule de Steiner pour les érosions (à paraître dans Adv. in Appl. Prob.).
I. Minkowski (1903):,Volumen und Oberfl.che. Math. Ann., Vol. 57, PP. 447–495.
R.T. Rockafellar (1972): Convex Analysis. Princeton University Press.
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© 1978 Springer-Verlag Berlin Heidelberg
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Matheron, G. (1978). The Infinitisimal Erosions. In: Miles, R.E., Serra, J. (eds) Geometrical Probability and Biological Structures: Buffon’s 200th Anniversary. Lecture Notes in Biomathematics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-93089-8_21
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DOI: https://doi.org/10.1007/978-3-642-93089-8_21
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