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Volume Product
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Volume Product
abstract
Our purpose here is to give an overview of known results and open questions concerning the volume product ${\mathcal P}(K)=\min_{z\in K}{\rm vol}(K){\rm vol}((K-z)^*)$ of a convex body $K$ in ${\mathbb R}^n$. We present a number of upper and lower bounds for ${\mathcal P}(K)$, in particular, we discuss the Mahler's conjecture on the lower bound of ${\mathcal P}(K)$, which is still open. We also show connections of ${\mathcal P}(K)$ with different parts of modern mathematics, including Geometric Number Theory, Convex Geometry, Analysis, Harmonic Analysis as well as Systolic and Symplectic Geometries and Probability.
Forward citations
Cited by 2 Pith papers
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The non-symmetric Mahler conjecture in dimension three
The paper proves the non-symmetric Mahler conjecture in dimension three by establishing the sharp lower bound of 64/9 for the non-symmetric volume product of any convex body in R^3.
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The non-symmetric Mahler conjecture in dimension three
The non-symmetric Mahler conjecture holds in dimension three: the volume product P(K) satisfies P(K) >= 64/9 for every convex body K in R^3.
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