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

A new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space ${\text{HH}}^n$ 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.