Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures Jianquan GeYinnian HeHaizhong LiHongfang Ma 53C4053A1052A20Alexandrov TheoremWulff shapeembedded hypersurface$r$-th anisotropic mean curvature Given a positive function $F$ on $S^{n}$ which satisfies a convexity condition, for $1 \leq r \leq n$, we define for hypersurfaces in $\mathbb{R}^{n+1}$ the $r$-th anisotropic mean curvature function $H^{F}_{r}$, a generalization of the usual $r$-th mean curvature function. We prove that a compact embedded hypersurface without boundary in $\mathbb{R}^{n+1}$ with $H^{F}_{r}$ constant is the Wulff shape, up to translations and homotheties. In the case $r = 1$, our result is the anisotropic version of Alexandrov's Theorem, which gives an affirmative answer to an open problem of F. Morgan. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3515 10.1512/iumj.2009.58.3515 en Indiana Univ. Math. J. 58 (2009) 853 - 868 state-of-the-art mathematics http://iumj.org/access/