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 inequali ty, the dual Brunn-Minkowski inequality, and the dual isoperimetric inequal ity. Two methods are used. One involves the application of strong forms of Clarkson's inequality for L-P norms that hold for nonnegative functions, an d the other utilizes a refinement of the arithmetic-geometric mean inequali ty. A new and more informative proof of the equivalence of the dual Brunn-M inkowski inequality and the dual Minkowski inequality is given. The main re sults are shown to be the best possible up to constant factors. (C) 1999 Ac ademic Press.