Self-dual polytope and self-dual smooth Wulff shape

For any Wulff shape $\mathcal{W}$, its dual Wulff shape and spherical Wulff shape $\widetilde{\mathcal{W}}$ can be defined naturally. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, we show that if a spherical convex polytope $P$ is of constant width $\delta$, then $\delta=\pi/2$. As an application of this fact, we prove that a polytope Wulff shape is self-dual if and only if its spherical Wulff shape is a spherical convex body of constant width. We also prove that a smooth Wulff shape is self-dual if and only if for any interior point $P$ of $\widetilde{\mathcal{W}}$ and for any point $Q$ of the intersection of the boundary of $\widetilde{\mathcal{W}}$ and the graph of its spherical support function (with respect to $P$), the image of $Q$ under the spherical blow-up (with respect to $P$) is always a boundary point of $\widetilde{\mathcal{W}}$.