Large parallel volumes of finite and compact sets in d-dimensional Euclidean space
M. Kiderlen18 Germany DK-8000 Aarhus C Tel. 0049 631 205 4825 Denmark
The -parallel volume of a compact subset in -dimensional Euclidean space is the volume of the set of all points of Euclidean distance at most from . According to Steiner's formula, is a polynomial in when is convex. For finite sets satisfying a certain geometric condition, a Laurent expansion of for large is obtained. The dependence of the coefficients on the geometry of is explicitly given by so-called intrinsic power volumes of . In the planar case such an expansion holds for all finite sets . Finally, when is a compact set in arbitrary dimension, it is shown that the difference of large -parallel volumes of and of its convex hull behaves like , where is an intrinsic power volume of .
Cite this article
J. Kampf, M. Kiderlen, Large parallel volumes of finite and compact sets in d-dimensional Euclidean space. Doc. Math. 18 (2013), pp. 275–295DOI 10.4171/DM/397