Isoperimetric Inequalities for affine and dual affine quermassintegrals

Abstract

In convex geometry, affine quermassintegrals are important quantities in the for studyof geometric inequalities. Lutwak introduced two types of these integrals: the dualaffine quermassintegrals, denoted by ˜Φk and the affine quermassintegrals, denoted byΦk. Of special interest are the isoperimetric inequalities associated with these integrals.For the dual affine quermassintegrals ˜Φk, this involves finding sharp upper bounds andidentifying the convex bodies of a given volume that achieve equality. For the affinequermassintegrals Φk, the goal is to establish sharp lower bounds and determine theconvex bodies that minimize them.The isoperimetric inequality for the dual affine quermassintegrals ˜Φk was proven earlier, with the inequality shown in [BS60] and the cases of equality discussed by Grinbergin [Gri91]. In contrast, the isoperimetric inequality for the affine quermassintegrals Φkremained an unsolved problem for many years. It was only in 2022 that this inequalitywas finally proven by E. Milman and Yehudayoff in [MY23].In this thesis, we give a self-contained presentation of the proofs of the isoperimetricinequalities for ˜Φk and Φk, establish the equality cases, and examine some of their prop-erties. Additionally, we will discuss some consequences of the isoperimetric inequalityand highlight important special cases that serve as fundamental tools in affine convexgeometry.