Abstract
Let for positive integersj,k,d and convex bodiesK of Euclideand-spaceE d of dimension at leastj V j, k (K) denote the maximum of the intrinsic volumesV j (C) of those convex bodies whosej-skeleton skel j (C) can be covered withk translates ofK. Then thej-thk-covering density θ j,k (K) is the ratiok V j (K)/V j,k (K). In particular, θ d,k refers to the case of covering the entire convex bodiesC and the density is measured with respect to the volume while forj=d-1 the surface of the bodiesC is covered and accordingly the density is measured with respect to the surface area.
The paper gives the estimate
for thej-thk-covering density and some related results.
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Gritzmann, P. Über diej-ten Überdeckungsdichten konvexer Körper. Monatshefte für Mathematik 103, 207–220 (1987). https://doi.org/10.1007/BF01364340
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DOI: https://doi.org/10.1007/BF01364340