Abstract: A bounded set in a normed vector space is diametrically complete if every strictly larger set has larger diameter. In Euclidean spaces, the diametrically complete sets are precisely the convex bodies of constant width, which have a rich literature. However, for n > 2, most (in the sense of Baire category) n-dimensional normed spaces have the property that their only bodies of constant width are the balls, whereas every bounded set has a (generally non-unique) completion, that is, a containing diametrically complete set of the same diameter. Thus, for normed spaces, diametrically complete sets are the proper counterpart to bodies of constant width. We discuss, mainly for finite dimensional spaces, some properties of diametrically complete sets and of the diametric completion mapping, which associates with every convex body the set of its completions. (Joint work with Jose Pedro Moreno)

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