Anisotropic curvature measures and uniqueness of convex bodies

We prove that an arbitrary convex body $C \subseteq \mathbf{R}^{n+1}$, whose $k$-th anisotropic curvature measure (for $k =0, \ldots , n-1$) is a multiple constant of the anisotropic perimeter of C, must be a rescaled and translated Wulff shape.This result provides a generalization of a theorem of Schneider (1979) and resolves a conjecture of Andrews and Wei (2017).