Geodesic
Large parallel volumes of finite and compact sets in d-dimensional Euclidean space
Documenta mathematica, Tome 18 (2013), pp. 275-295
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The r-parallel volume V(Cr​) of a compact subset C in d-di­men­sional Euclidean space is the volume of the set Cr​ of all points of Euclidean distance at most r>0 from C. According to Steiner's formula, V(Cr​) is a polynomial in r when C is convex. For finite sets C satisfying a certain geometric condition, a Laurent expansion of V(Cr​) for large r is obtained. The dependence of the coefficients on the geometry of C is explicitly given by so-called intrinsic power volumes of C. In the planar case such an expansion holds for all finite sets C. Finally, when C is a compact set in arbitrary dimension, it is shown that the difference of large r-parallel volumes of C and of its convex hull behaves like crd−3, where c is an intrinsic power volume of C.
DOI : 10.4171/dm/397
Classification : 41A10, 52A39, 52B11
Mots-clés : large parallel sets, Laurent expansion of parallel volume, Steiner formula, intrinsic power volume 
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     author = {J. Kampf and M. Kiderlen},
     title = {Large parallel volumes of finite and compact sets in d-dimensional {Euclidean} space},
     journal = {Documenta mathematica},
     pages = {275--295},
     year = {2013},
     volume = {18},
     doi = {10.4171/dm/397},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/397/}
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J. Kampf; M. Kiderlen. Large parallel volumes of finite and compact sets in d-dimensional Euclidean space. Documenta mathematica, Tome 18 (2013), pp. 275-295. doi: 10.4171/dm/397

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