Empirical Forms of the Petty Projection Inequality for Random Convex Sets
By re-examining the Petty projection inequality from a stochastic viewpoint, this work establishes empirical (random) versions of it: sharp extremal inequalities for multiple-entry functionals of random convex sets, including mixed projection bodies and mixed volumes. The classical Petty projection inequality is recovered as a limiting case of these random approximations, and the method unifies related affine isoperimetric inequalities through the lens of convex bodies generated by independent random points.
2501.00253
The Petty projection inequality is an affine isoperimetric inequality in convex geometry stating that, among convex bodies of a given volume, ellipsoids maximize the volume of the polar projection bo…