  We aim to study the classical Rosenthal-Szasz inequality for a plane whose
geometry is given by a norm. This inequality states that the bodies of constant
width have the largest perimeter among all planar convex bodies of given
diameter. In the case where the unit circle of the norm is given by a Radon
curve, we obtain an inequality which is completely analogous to the Euclidean
case. For arbitrary norms we obtain an upper bound for the perimeter calculated
in the anti-norm, yielding an analogous characterization of all curves of
constant width. To derive these results, we use methods from the differential
geometry of curves in normed planes.
