On Minkowski Bodies of Constant Width

A metric set is entire if the addition of any point to the set increases the diameter. A convex body has constant width if all pairs of parallel supporting planes are the same distance apart. These concepts are known to be equivalent in euclidean space. The present paper shows that they are also equivalent in a minkowski space. A proof for this equivalence for the minkowski plane was given by Meissner. He showed also that two curves of the same constant width have the same circumference, and that a three-dimensional body of constant width has plane sections of constant width, in terms of the corresponding section of the minkowski sphere as plane indicatrix. However, Meissner's three-space proof for the equivalence of entireness to constant width is incomplete. He assumed, moreover, that the indicatrix had no singular points. The equivalence here is shown for the w-dimensional case with the assumption merely that the indicatrix is convex. In a euclidean E space let C be a closed, convex hypersurface with 0 as center. In terms of C as indicatrix let a minkowski distance be defined in the usual way to make the space an M. Let r and s be half-rays emanating from 0 at an angle 0 and let Xr, X, be the euclidean lengths of the minkowski radii of C in these directions. Then sm(r, s), a positional sine with respect to C, is defined to mean XrX, sin 0. 5