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On two finite covering problems of Bambah, Rogers, Woods and Zassenhaus. (English) Zbl 0564.52008
R. P. Bambah, C. A. Rogers, A. C. Woods, and H. Zassenhaus [J. Lond. Math. Soc., II. Ser. 27, 304–314 (1952; Zbl 0046.38004), Acta Arith. 9, 191–207 (1964; Zbl 0127.27602), Monatsh. Math. 72, 107–117 (1968; Zbl 0169.24603), Mathematika 18, 91–97 (1971; Zbl 0228.52005)] considered the general problem of covering planar convex bodies \(C\) by \(k\) translates of a centrally-symmetric convex body \(K\) of \(E^ 2\) with the ramification that these translates cover the convex hull \(C_ k\) of their centres. They proved interesting inequalities for the volume of \(C\) and \(C_ k\).
In the present paper some analogous results in Euclidean \(d\)-space \(E^ d\) are given. It turns out that on the one hand extremal configurations for \(d\geq 5\) are of quite different type than in the planar case. On the other hand inequalities similar to the planar ones seem to exist in general. Inequalities in both directions for the volume and other quermassintegrals are given.
Reviewer: P. Gritzmann

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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