Valuations on convex sets, non-commutative determinants, and pluripotential theory

Abstract

A new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space H n is presented. In particular new examples of Sp ( n ) Sp ( 1 ) -invariant translation invariant continuous valuations are constructed. This method is based on the theory of plurisubharmonic functions of quaternionic variables developed by the author in two previous papers (Bull. Sci. Math. 127 (1) (2003) 1; J. Geom. Anal. 13 (2) (2003) 183).