The intrinsic volumes (AKA Minkowski Functionals) $V_i:K\to \mathbb{R}$ of compact, convex subsets of $\mathbb{R}^d$ (denoted $K$) are important valuations in convex geometry.
My question is simply whether these $V_i$ are defined for anything beyond polyconvex, compact sets.
I have found many sources which define them for polyconvex and compact sets, but none which go beyond that. For example, can the function be meaningfully extended to open sets or to 2D surfaces with boundary embedded in $\mathbb{R}^3$? Any knowledge of an extension beyond polyconvex, compact sets is appreciated!