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 $L^p$ 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