Pluripotential construction of convex-body valuations

Statement

Let $K\mapsto h_K$ be the support-function embedding of convex bodies. Integrating a compactly supported test object against a fixed-degree complex or quaternionic Hessian measure of $h_K$ defines a continuous translation-invariant valuation.

In the quaternionic case, for suitable compactly supported data $V^{(1)},\ldots,V^{(n-k)}$ and $\psi$ , a representative family is

K\longmapsto \int_{(\mathbb H^n)^*} D\!\left((\operatorname{Hess}_{\mathbb H}h_K)[k], V^{(1)},\ldots,V^{(n-k)}\right)\psi\,dV.

Why it is continuous

Hausdorff convergence $K_m\to K$ gives locally uniform convergence $h_{K_m}\to h_K$ . The continuity of quaternionic Monge–Ampère measures then gives convergence of the displayed integrals.

Why it is a valuation

When $K\cup L$ is convex, support functions convert union and intersection into maximum and minimum. The quaternionic Błocki formula converts those max/min operations into inclusion–exclusion for the Hessian measures. Translation adds a linear function to $h_K$ , which its Hessian annihilates.

Complex, quaternionic, and octonionic branches

The complex construction uses powers of $dd^ch_K$ paired with compactly supported forms. The quaternionic construction uses mixed Moore determinants. On $\mathbb O^2$ , the analogous determinant measure produces the octonionic pseudovolume. These are parallel mechanisms with different underlying Hessians.

References

Semyon Alesker, “Valuations on convex sets, non-commutative determinants, and pluripotential theory,” Advances in Mathematics 195 (2005), 561–595. arXiv record. Relevant: Theorems 4.1.3 and 4.2.1.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: §4.