Mahler's volume product conjecture

Let $(X,\|\cdot\|)$ be an $n$-dimensional normed space, let $(X^*,\|\cdot\|^*)$ be its dual, and let $B$ and $B^{\circ}$ be the unit balls of $X$ and $X^*$, respectively. Write $\nu(X)$ for the Liouville volume of $B\times B^{\circ}\subset X\times X^*$, also called the Mahler volume or volume product. Mahler's volume product conjecture. For every $n$-dimensional normed space $X$,

$$\nu(X)\geq\frac{4^n}{n!}.$$

The conjecture concerns the sharp lower bound for the volume product of centrally symmetric convex bodies, with equality attained by the hypercube and, non-uniquely in the broad formulation, its relevant affine counterparts. It was conjectured by Mahler in 1939 and remains an open problem in convex geometry.