Geodesic
Asymptotic Order of the Parallel Volume Difference in Minkowski Spaces
Journal of convex analysis, Tome 21 (2014) no. 4, pp. 925-95

Voir la notice de l'article provenant de la source Heldermann Verlag

We investigate the asymptotic behavior of the parallel volume of fixed non-convex bodies in Minkowski spaces as the distance $r$ tends to infinity. We will show that the difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself, which is called parallel volume difference, can at most have order $r^{d-2}$ in a $d$-dimensional Minkowski space. Then we will show that in certain Minkowski spaces (and in particular in Euclidean spaces) this difference can at most have order $r^{d-3}$. We will characterize the $2$-dimensional Minkowski spaces in which the parallel volume difference has always at most order $r^{-1}$. Finally we present applications concerning Brownian paths and Boolean models.
Classification : 52A20, 52A21, 52A22, 52A38
Mots-clés : Convex geometry, parallel volume, non-convex body, random body 
@article{JCA_2014_21_4_JCA_2014_21_4_a1,
     author = {J. Kampf},
     title = {Asymptotic {Order} of the {Parallel} {Volume} {Difference} in {Minkowski} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {925--95},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2014},
     url = {http://geodesic.mathdoc.fr/item/JCA_2014_21_4_JCA_2014_21_4_a1/}
}
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J. Kampf. Asymptotic Order of the Parallel Volume Difference in Minkowski Spaces. Journal of convex analysis, Tome 21 (2014) no. 4, pp. 925-95. http://geodesic.mathdoc.fr/item/JCA_2014_21_4_JCA_2014_21_4_a1/