ON MAHLER COMPOUND BODIES

Let 1 less-than-or-equal-to M less-than-or-equal-to N - 1 be integers and K be a convex, symmetric set in Euclidean N-space. Associated with K and M, Mahler identified the M(th) compound body of K, [K]M, in Euc lidean (M/N)-space. The compound body [K]M is describable as the conve x hull of a certain subset of the Grassmann manifold in Euclidean (M/N )-space determined by K and M. The sets K and [K]M are related by a nu mber of well-known inequalities due to Mahler. Here we generalize this theory to the geometry of numbers over the adele ring of a number fie ld and prove theorems which compare an adelic set with its adelic comp ound body. In addition, we include a comparison of the adelic compound body with the adelic polar body and prove an adelic general transfer principle which has implications to Diophantine approximation over num ber fields.