Stability of Inequalities in the Dual Brunn-Minkowski Theory*

Stability versions are given of several inequalities from E. Lutwak’s dual Brunn- Minkowski theory. These include the dual Aleksandrov-Fenchel inequality, the dual Brunn-Minkowski inequality, and the dual isoperimetric inequality. Two methods are used. One involves the application of strong forms of Clarkson’s inequality for Lp norms that hold for nonnegative functions, and the other utilizes a refinement of the arithmetic-geometric mean inequality. A new and more informative proof of the equivalence of the dual Brunn-Minkowski inequality and the dual Minkowski inequality is given. The main results are shown to be the best possible up to constant factors.