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.
Categories: math.MG math.FA