Quaternionic Błocki formula

Statement

Let $u,v$ be continuous quaternionic plurisubharmonic functions on $\Omega\subseteq\mathbb H^n$ , and assume that $\min\{u,v\}$ is also quaternionic PSH. Then

\operatorname{MA}_{\mathbb H}(\max\{u,v\}) +\operatorname{MA}_{\mathbb H}(\min\{u,v\}) =\operatorname{MA}_{\mathbb H}(u)+\operatorname{MA}_{\mathbb H}(v).

There are corresponding polarized identities for mixed quaternionic Monge–Ampère measures.

Why the minimum is an hypothesis

The maximum of two quaternionic PSH functions is again quaternionic PSH, but their minimum need not be. The displayed identity therefore cannot be stated for an arbitrary pair without the extra assumption.

Role in valuation theory

For support functions of convex bodies, $h_{K\cup L}=\max(h_K,h_L)$ when $K\cup L$ is convex, and $h_{K\cap L}=\min(h_K,h_L)$ . The formula turns this max/min behavior into the inclusion–exclusion identity for a valuation on convex bodies.

References

Zbigniew Błocki, “On the definition of the Monge–Ampère operator in $\mathbb C^2$ ,” Mathematische Annalen 328 (2004), 415–423. DOI record.
Semyon Alesker, “Valuations on convex sets, non-commutative determinants, and pluripotential theory,” Advances in Mathematics 195 (2005), 561–595. arXiv record. Relevant: Theorem 3.2.1.