Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets

TL;DR

This paper introduces a new class of plurisubharmonic functions on the octonionic plane, extending classical theorems and constructing novel Spin(9)-invariant valuations on convex sets in 16-dimensional space.

Contribution

It develops an octonionic analogue of key theorems and constructs new continuous, translation-invariant valuations, including a Spin(9)-invariant valuation on R^{16}.

Findings

Established an octonionic version of classical theorems.

Constructed new examples of valuations on convex sets.

Provided a novel Spin(9)-invariant valuation on R^{16}.

Abstract

A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.