A Characterization of Curves of Constant Width

Abstract

This contradicts the assumption k = 27r, since the quantity on the left of (11) is no larger that k2/4 = 7r2 (square formula 5). Hence equalities (6), (7), and (8) hold. These equalities are true of any selection xo, x1, *., x,, (as described). We conclude from (8) that the diameters of C have the same midpoint M; from (7) that they have the same length (which must be 2); hence the points of C are all on the sphere of radius 1 about M. The only path on this sphere, from xo to its antipode A(xo), having length ir, is half of a great circle; so formula (6) shows C must consist of two great semi-circles, which in fact form a single great circle because the diameters all pass through M. This verifies the second conjecture.