Self-dual Polytope and Self-dual Smooth Wulff Shape

For any Wulff shape W, its dual Wulff shape and spherical Wulff shape \(\widetilde{W}\) $\tilde{W}$ can be defined naturally. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, we prove that a polytope 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 points P of \(\widetilde{W}\) $\tilde{W}$ and for any point Q of the intersection of the boundary of \(\widetilde{W}\) $\tilde{W}$ 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 point of \(\widetilde{W}\) $\tilde{W}$ . Moreover, we give an affirmative answer to the problem posed by M. Lassak which says that “Do there exist reduced spherical n-dimensional polytopes (possibly some simplices?) on \(\mathbb {S}^n\) ${\mathbb{S}}^n$ , where \(n\ge 3\) $n\ge 3$ , different from the \(1/2^n\) $1/2^n$ part of \(\mathbb {S}^n?\) ${\mathbb{S}}^n?$ ”.