Octonionic pseudovolume

Core idea

Let $K\subseteq\mathbb O^2\cong\mathbb R^{16}$ be a convex body, let $h_K$ be its support function, and let $B$ be the centered unit ball. The octonionic pseudovolume is

P_{\mathbb O}(K)= \int_B \det\!\left(\operatorname{Hess}_{\mathbb O}h_K\right)dq,

with the Hessian determinant interpreted as the octonionic Monge–Ampère measure.

Valuation and invariance

The functional $P_{\mathbb O}$ is a continuous translation-invariant valuation and is invariant under the $\operatorname{Spin}(9)$ spin action on $\mathbb O^2$ .

Mechanism

Continuity follows because Hausdorff convergence of convex bodies gives locally uniform convergence of support functions and hence weak convergence of their octonionic Hessian measures. The valuation identity follows from the max-min formula for those measures. Translation invariance follows because translating $K$ adds a linear function to $h_K$ , leaving its Hessian unchanged.

General test functions

More generally, integrating the Hessian measure against any compactly supported continuous test function gives a continuous translation-invariant valuation. Choosing the radial test function $1_B$ exposes the full $\operatorname{Spin}(9)$ symmetry and yields $P_{\mathbb O}$ .

References

Semyon Alesker, “Plurisubharmonic functions on the octonionic plane and $\operatorname{Spin}(9)$ -invariant valuations on convex sets,” Journal of Geometric Analysis 18 (2008), 651–686. arXiv record. Relevant: Theorem 0.1.8 and §4.