J Austral Math Soc Ser A 55 pp183--215, 1993.

On Mahler's Compound Bodies

Edward B. Burger

Abstract

Let 1 £ M £ 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, áKñ_M, in Euclidean (^N_M)-space. The compound body áKñ_M is describable as the convex hull of a certain subset of the Grassmann manifold in Euclidean (^N_M)-space. The sets K and áKñ_M are related by a number of well-know inequalities due to Mahler. Here we generalize this theory to the geometry of numbers over the adele ring of a number field and prove theorems which compare an adelic set with its adelic polar body. In addition, we include a comparison of the adelic compound body with the adelic polar and prove an adelic general transfer principle which has implications to Diophantine approximation over number fields.

1991 AMS Subject Classification: primary 11H06, 11R56; secondary 11J13, 11J61

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Authors

Edward J. Burger

Williams College, Williamstown, Massachusetts 01267, USA.