Blaschke–Santaló inequality

In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is known that the shapes with the largest possible Mahler volume are the balls and solid ellipsoids; this is now known as the Blaschke–Santaló inequality. The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube.

A convex body in Euclidean space is defined as a compact convex set with non-empty interior. If B is a centrally symmetric convex body in n-dimensional Euclidean space, the polar body B^o is another centrally symmetric body in the same space, defined as the set

The Mahler volume of B is the product of the volumes of B and B^o.

If T is an invertible linear transformation, then ${\displaystyle (TB)^{\circ }=(T^{-1})^{\ast }B^{\circ }}$; thus applying T to B changes its volume by ${\displaystyle \det T}$ and changes the volume of B^o by ${\displaystyle \det(T^{-1})^{\ast }}$. Thus the overall Mahler volume of B is preserved by linear transformations.

...
Wikipedia