The Brunn]Minkowski Inequality, Minkowski’s First Inequality, and Their Duals1

Quantitative versions are given of the equivalence of the Brunn]Minkowski inequality and Minkowski’s first inequality from the Brunn]Minkowski theory. Similar quantitative versions are obtained of the equivalence of the corresponding inequalities from Lutwak’s dual Brunn]Minkowski theory. The main results are shown to be the best possible up to constant factors