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This article concerns packings and coverings that are formed by the application of rigid motions to the members of a given collection K of convex bodies. There are two possibilities to construct such packings and coverings: One may permit that the convex bodies from K are used repeatedly, or one may require that these bodies should be used at most once. In(More)
It is well known that an n-dimensional convex body permits a lattice packing of density I only if it is a centrally symmetric potytope of at most 2(2" 1) facets. This article concerns itself with the associated stability problem whether a convex body that permits a packing of high density is in some sense close to such a polytope. Several inequalities that(More)
f(X) è 1 is called a star body (cf. K. Mahler [l]). The function by which a star body is defined is determined uniquely. Suppose now that 5 is bounded. There exists a least number k such that for all X and Y f(X+ Y) èHf(X)+f(Y)); this number k is called the concavity coefficient of 5. One has always k ^ 1 and furthermore k = 1 if and only if 5 is convex.(More)
With any given convex body we associate three numbers that exhibit, respectively, its deviation from a ball, a centrally symmetric body, and a body of constant width. Several properties of these deviation measures are studied. Then, noting that these special bodies may be defined in terms of their normals, corresponding deviation measures for normals are(More)
Let [Kx, K2, ■ ■ ■ } be a class of compact convex subsets of euclidean M-space with the property that the set of their diameters is bounded. It is shown that the sets A, can be rearranged by the application of rigid motions so as to cover the total space if and only if the sum of the volumes of all the sets A, is infinite. Also, some statements regarding(More)
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