Subspace concentration of zonoids and a sharp Minkowski mixed volume inequality

The mixed volume measure $V_{K_1,\dots ,K_n}$ of compact convex sets $K_1,\dots ,K_n$ in $R^n$ is the localization of the classic Minkowski mixed volume $V(K_1,\dots ,K_n)$ and is one of the generalizations of the important cone-volume measure. When $K_1,\dots ,K_{n-1}$ are zonotopes and $K_n$ is a convex body or $K_1,\dots ,K_{n-1},K_n$ are zonoids in $R^n$, the subspace concentration of $V_{K_1,\dots ,K_n}$ is proved. As applications, a subspace concentration phenomenon for quermassintegrals is revealed and a sharp affine isoperimetric inequality for the Minkowski mixed volume is established.