There is an important number associated with convex bodies called the volume product. One reason that it is important is that it is invariant under invertible linear transformations. The precise upper bound for this number is known in all dimensions. In the late 1930's, Mahler gave the precise lower bound in dimension two and conjectured the value in higher dimensions but, in the almost 70 years since, the conjecture has not been proved.
In the talk, I will give the background to the inequality, prove Mahler's two dimensional result and say what is known in higher dimensions.
