Surfaces of generalized constant width

An orientable closed connected surface $\Phi$ is called a surface of generalized constant width $d$ if: 1) the end of the vector $Op^*=Op+dn(p)$ lies on $\Phi$ for every $p\in\Phi$, where $n(p)$ is the inward unit normal; 2) the map $\varphi\colon p\to p^*$ is an involution. We prove the following
Theorem. If $\Phi$ is an analytic surface of generalized constant width $d$ and satisfies the condition $|K(p)|=|K(p^*)|$ then $\Phi$ is a sphere, with $K(p)$ denoting the Gaussian curvature of $\Phi$ at $p$.