A characteristic property of the ball

In 1926 Nakajima showed that any convex body in $\mathbb{R}^3$ with constant width, constant brightness, and boundary of class $C^2$ is a ball [8]. To prove this Nakajima used the existence of umbilic point on every closed convex surfaces, topological theorem about a continuous tangent vector field on the sphere $S^2$ and isoperimetric inequality. The alternative proof of Nakajima Theorem under the same restrictions on the boundary of a convex body is given in this article. We reduce the Nakajima problem to the Monge–Ampere equation for the support function. We claim that the right part of this equation don't be negative. The last statement is proved with theorems about the structure for surfaces of negative curvature. Then we show that every orthogonal projection of body onto the plane is a circle.