The Infinitisimal Erosions

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 S_A of the surface measure associated with A only. More precisely: K(A) is the largest convex on S_A. 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).