The Busemann intersection inequality is a fundamental isoperimetric principle for the volume of slices of a convex body. It is closely connected to boundedness of the Radon and k-plane transforms.  

The original proof of the Busemann intersection inequality relied on symmetrization but rather indirectly, as monotonicity under a single symmetrization was not addressed.  More recent proofs by Adamczak--Paouris--Pivovarov--Simanjuntak and Milman--Yehudayoff--Shabelman explicitly treat such monotonicity.    

I will discuss a new form of the Busemann intersection inequality. It avoids working with slices, and instead relies on enumerating random points in slabs.  I will explain how this "empirical" version of the Busemann intersection inequality is well-suited to direct analysis under symmetrization.  

Based on joint work with  G. Paouris (Texas A\&M) and P. Simanjuntak (Texas A\&M).