Uniqueness of the surface energy density for a Wulff shape with $C^1$ boundary

Let $\gamma :S^n\to {\mathbb{R}}_{+}$ be a continuous function and let ${\mathcal{W}}_{\gamma }$ be the Wulff shape associated with $\gamma$. In this paper, we show that if the boundary of ${\mathcal{W}}_{\gamma }$ is a $C^1$ submanifold, then $\gamma$ must be the convex integrand of ${\mathcal{W}}_{\gamma }$.