Self-dual Polytope and Self-dual Smooth Wulff Shape

Abstract

For any Wulff shape W, its dual Wulff shape and spherical Wulff shape $\widetilde{W}$ 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 $\widetilde{W}$ and for any point Q of the intersection of the boundary of $\widetilde{W}$ 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 $\widetilde{W}$. 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 $\mathbb {S}^n$, where $n\ge 3$, different from the $1/2^n$ part of $\mathbb {S}^n?$”.