# Intrinsic volumes of non-polyconvex, non-compact sets

I am reposting this question I asked and bountied on Math SE, which has been upvoted but not answered or commented on.

The intrinsic volumes (AKA Minkowski Functionals or, with different normalization, quermassintrgrals) $$V_i:K\to \mathbb{R}$$ of compact, convex subsets of $$\mathbb{R}^d$$ (denoted $$K$$) are important valuations in convex geometry. Hadwiger's Theorem states that, up to linear combination, these are the only valuations on compact, convex subsets of $$\mathbb{R}^d$$. These can be further extended to sets which are polyconvex, as shown in several sources, including Daniel A. Klain and Gian-Carlo Rota's Introduction to Geometric Probability.

My question is simply whether these intrinsic volumes $$V_i$$ are known to be 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!

For a full overview of intrinsic volumes, see "Introduction to Geometric Probability" by Klain and Rota.

• Quermassintegrals can be defined for arbitrary smooth open subsets of $R^d$ (actually only $C^2$-regularity is necessary). More precisely, they can be expressed as integrals of appropriate functions of the (extrinsic) curvature tensor of the boundary. See, for example, the introduction of this paper by Trudinger (formula (1.3)). Nov 5, 2019 at 17:20

The largest level of generality for which intrinsic volumes can be defined is a difficult question. One large class of compact sets for which they can be defined is the class of compact sets admitting a normal cycle. How large this class is is not very well understood, but it contains smooth (or more generally positive reach), polyconvex, and subanalytic sets for example.

• Could you post a reference which talks about the intrinsic volumes of sets admitting a normal cycle? Also, does that mean that the largest class on which intrinsic volumes can be defined is an open question? Nov 5, 2019 at 17:30
• sciencedirect.com/science/article/pii/S0926224503000202 is one possible reference. And yes, the largest class for which intrinsic volumes can be defined while retaining their basic properties is open as far as I know. In fact, it is not even clear that it exists. Maybe there are many maximal classes but no single one including them all, depending on the formulation of the problem. Nov 5, 2019 at 17:42