Convex Bodies of Constant Width and Constant Brightness

In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in ${\mathbb{R}}^3$ with constant width, constant brightness, and boundary of class $C^2$ is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.