Valuation on convex bodies

Summary

A valuation on convex bodies is a function $\varphi$ on the nonempty compact convex sets of $\R^n$ that is additive in the inclusion–exclusion sense. This means $\varphi(K \cup L) + \varphi(K \cap L) = \varphi(K) + \varphi(L)$ whenever $K \cup L$ is convex. Volume, surface area, the Euler characteristic and all intrinsic volumes are valuations. Valuations are the finitely additive "measures" of convex geometry, and their classification, starting with Hadwiger's theorem, is a central theme of integral geometry. This notion is unrelated to valuations of fields in algebra.