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

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.