Valuation on convex bodies

Definition

Let $\mathcal K(V)$ be the family of convex bodies in a finite-dimensional real vector space. A function $\phi:\mathcal K(V)\to A$ , with values in an abelian group $A$ , is a valuation if

\phi(K\cup L)+\phi(K\cap L)=\phi(K)+\phi(L)

whenever $K,L,K\cup L\in\mathcal K(V)$ .

Regularity and invariance

A real- or complex-valued valuation is continuous if it is continuous for the Hausdorff metric. It is translation-invariant if $\phi(K+x)=\phi(K)$ for every $x\in V$ , and $G$ -invariant for a linear group $G$ if $\phi(gK)=\phi(K)$ for every $g\in G$ . These are additional predicates, not part of finite additivity.

Examples

Euclidean volume and the constant Euler-characteristic valuation $\chi(K)=1$ are continuous and translation-invariant. Fixing all but one argument of a mixed volume gives another standard family. Pluripotential theory supplies further examples by applying Hessian measures to support functions.

Group-invariant theory

If a compact subgroup $G\subseteq O(V)$ acts transitively on the unit sphere, the space of continuous translation- and $G$ -invariant valuations is finite dimensional. Quaternionic groups and the exceptional group $\operatorname{Spin}(9)$ enter valuation theory through this principle.

References

Peter McMullen and Rolf Schneider, “Valuations on convex bodies,” in Convexity and Its Applications, Birkhäuser, 1983, 170–247. DOI record.
Semyon Alesker, “Valuations on convex sets, non-commutative determinants, and pluripotential theory,” Advances in Mathematics 195 (2005), 561–595. arXiv record.