Let $E,M$ be two convex subsets of $\mathbb{R}^n$. Then the Alexandrov-Fenchel theorem states that the coefficients $V_j(E,M)$ in $$\mathrm{Vol}(xE+yM)=\sum\limits_{j}{n\choose j}V_j(E,M)x^{n-j}y^j$$ satisfy $V_j(E,M)^2\geq V_{j-1}(E,M)V_{j+1}(E,M)$.

Is there a version of this for Riemannian manifolds? That if $E,M$ are subsets of a Riemannian manifold, then the same relation holds? Any references would be great.