Volume Product

Our purpose here is to give an overview of known results and open questions concerning the volume product $\mathcal{P}(K)={\min }_{z\in K}\mathrm{v}\mathrm{o}\mathrm{l}(K)\mathrm{v}\mathrm{o}\mathrm{l}((K-z{)}^{\ast })$ 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.