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Über die j-ten Überdeckungsdichten konvexer Körper. (On the j-th covering densities of convex bodies). (German) Zbl 0624.52006

The present paper deals with the following finite covering problem: For a given convex body K in Euclidean d-space \(E^ d\) and integers j, k with \(0\leq j\leq d\), \(1\leq k\) determine the maximum \(V_{j,k}(K)\) of the intrinsic volume \(V_ j\) of convex bodies C whose j-skeleton can be covered by k translates of K. (Recall that \(V_ d\) is the usual volume, \(2V_{d-1}\) is the surface area etc.)
Denoting for dim \(K\geq j\) the j-th k-covering density \(kV_ j(K)V^{- 1}_{j,k}(K)\) of K with \(\vartheta_{j,k}(K)\), we particularly prove the inequality \[ 1\leq \vartheta_{j,k}(K)< e(j+\sqrt{\pi /2}\sqrt{d- j})< e(d+1) \] and give some bounds for a related problem.

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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References:

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