This paper introduces the \textit{ anisotropic $\omega_0$-capillary $p$-sum} of two hypersurfaces in $\mathbb{ R}_+^{ n+1}$, and establishes a theory for anisotropic capillary convex bodies. For a smooth convex hypersurface $\Sigma $ with anisotropic $\omega_0$-capillary boundary, we compute the variation of its anisotropic capillary $k$-th quermassintegral via this $p$-sum, thereby defining the associated anisotropic $\omega_0$-capillary $k$-th $p$-surface area measure on the capillary Wulff shape $\mathcal{ C}_{ \omega_{ 0}}$. This motivates us to propose and solve the anisotropic capillary $L_{ p}$-Minkowski problem for $p\geq1$.
Categories: math.DG