Large parallel volumes of finite and compact sets in d-dimensional Euclidean space

  • J. Kampf

  • M. Kiderlen

    18 Germany DK-8000 Aarhus C Tel. 0049 631 205 4825 Denmark
Large parallel volumes of finite and compact sets in d-dimensional Euclidean space cover
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Abstract

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. 275–295

DOI 10.4171/DM/397