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 rr-parallel volume V(Cr)V(C_r) of a compact subset CC in dd-di­men­sional Euclidean space is the volume of the set CrC_r of all points of Euclidean distance at most r>0r>0 from CC. According to Steiner's formula, V(Cr)V(C_r) is a polynomial in rr when CC is convex. For finite sets CC satisfying a certain geometric condition, a Laurent expansion of V(Cr)V(C_r) for large rr is obtained. The dependence of the coefficients on the geometry of CC is explicitly given by so-called intrinsic power volumes of CC. In the planar case such an expansion holds for all finite sets CC. Finally, when CC is a compact set in arbitrary dimension, it is shown that the difference of large rr-parallel volumes of CC and of its convex hull behaves like crd3cr^{d-3}, where cc is an intrinsic power volume of CC.

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