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Visibility and diameter maximization of convex bodies

Abstract This paper considers the problem of diameter maximization of a convex body in the Euclidean space and its connection with Valentine's notion of visibility. The so-called diametral spectrum, or set of critical lengths, is studied in depth from a topological point of view. As a side result, we give a new characterization of planar convex bodies of constant width. Our methodology emphasizes the role played by the concepts of critical pair and critical curve.