Valuation (geometry)

In geometry, a valuation is a finitely additive function from a collection of subsets of a set ${\displaystyle X}$ to an abelian semigroup. For example, Lebesgue measure is a valuation on finite unions of convex bodies of ${\displaystyle \mathbb {R} ^{n}.}$ Other examples of valuations on finite unions of convex bodies of ${\displaystyle \mathbb {R} ^{n}}$ are surface area, mean width, and Euler characteristic.

In geometry, continuity (or smoothness) conditions are often imposed on valuations, but there are also purely discrete facets of the theory. In fact, the concept of valuation has its origin in the dissection theory of polytopes and in particular Hilbert's third problem, which has grown into a rich theory reliant on tools from abstract algebra.