Stability of inequalities in the dual Brunn-Minkowski theory
The dual Brunn-Minkowski theory, initiated by Erwin Lutwak, has a series of inequalities which can be considered as dual counterparts to classical inequalities on mixed volumes and related functionals. A central topic of the dual theory are inequalities for integrals involving radial functions of star bodies. In the present paper, the authors obtain strong stability versions of some of these inequalities. These include the dual Brunn-Minkowski inequality, the dual Aleksandrov-Fenchel inequality, and the dual isoperimetric inequality. The proofs involve, among other tools, strong forms of Clarkson's inequality for \(L^p\) norms, and a refinement of the arithmetic-geometric mean inequality due to Kober. Finally, the authors give a more informative proof for the equivalence of the dual Brunn-Minkowski inequality and the dual Minkowski inequality.
scientific article; zbMATH DE number 1274907
scientific article