Abstract
1. Let En be the n-dimensional Euclidean space and let K be a bounded closed convex set in En. For any direction u there are exactly two supporting hyperplanes of K, orthogonal to u, and the distance D,(K) between them is called the width of K in the direction u. By a hyperplane we always mean one of dimension n -1. K is called a set of constant width 1, or briefly of width 1, if SD(K)=1 for all u. A sphere S,(x) denotes the closed solid sphere in En of radius r about the center x. S,(x) is the circum-sphere (the in-sphere) of K if KCS,(x)(Sr(x)CK) and the radius r is smallest (largest) possible. We shall prove the following
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