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Post Closed as "Needs details or clarity" by M. Winter, Max Horn, Mikhail Katz, user44191, Carl-Fredrik Nyberg Brodda
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Michael Hardy
Michael Hardy
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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 $$\mathrm{Vol}(xE+yM)=\sum\limits_{j}{n\choose j}V_j(E,M)x^{n-j}y^j$$$$\operatorname{Vol}(xE+yM)=\sum_j {n\choose j} V_j(E,M)x^{n-j}y^j$$ 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.

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.

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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matilda
matilda
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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(n,j)$$V_j(E,M)$ in $$\mathrm{Vol}(xE+yM)=\sum\limits_{j}{n\choose j}V(n,j)x^{n-j}y^j$$$$\mathrm{Vol}(xE+yM)=\sum\limits_{j}{n\choose j}V_j(E,M)x^{n-j}y^j$$ satisfy $V(n,j)^2\geq V(n,j-1)V(n,j+1)$$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.

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.

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.

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matilda
matilda
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Alexandrov-Fenchel theorem for Riemannian manifolds

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.