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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 $$\operatorname{Vol}(xE+yM)=\sum_j {n\choose j} V_j(E,M)x^{n-j}y^j$$ 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.

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    $\begingroup$ In an arbitrary riemannian manifold, it is not clear what a linear combination xE + yM of two subsets E, M means. Maybe you want to restrict your question to cases where that makes sense. $\endgroup$
    – Daniel Asimov
    Commented Aug 18, 2023 at 3:29
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    $\begingroup$ @DanielAsimov Perhaps one possible sense could be the following: We change the metric on $\mathbb{R}^n$ and compute the volum wrt the new metric. The convexity can have two interpretation: wrt the straight lines or wrt the geodesics of metrics. Is it a reasonable sense for consideration of the generalized Fenchel theorem? $\endgroup$
    – Ali Taghavi
    Commented Sep 13, 2023 at 5:44

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