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 $\mathcal{W}_\gamma$.