Rigid motion invariant valuations on polytopes

Abstract: We show that any measurable, translation and $\mathrm{SO}(n)$-invariant valuation on polytopes in $\mathbb{R}<sup>n$ is a linear combination of the intrinsic volumes, which extends Hadwiger's classical characterization of rigid motion invariant continuous valuations on convex bodies. This result is based on a novel regularity result for translation invariant, measurable, $1$-homogeneous, and simple valuations. As a consequence, we obtain a similar characterization of the Steiner point map on polytopes.

• The paper proves that every measurable, translation- and rotation-invariant valuation on polytopes is a linear combination of the intrinsic volumes, extending Hadwiger’s theorem beyond continuous valuations.
• It replaces averaging convex bodies with averaging an explicit Stiefel-manifold representation, using equivariant regularity to establish the result and characterize the Steiner point as the unique measurable Minkowski-additive equivariant map.
• The results include the polytopal characterization of volume for simple valuations, while locally bounded and nonnegative valuation cases remain open research problems.

Overview and main results

This paper by Jonas Knoerr resolves a long-standing question posed by McMullen and Schneider in their 1983 survey on valuations on convex bodies: whether Hadwiger's characterization of rigid motion invariant continuous valuations remains valid when the valuation is defined only on the class of convex polytopes $\mathcal{P}^n \subset \mathcal{K}^n$. The paper answers affirmatively, and strengthens the regularity hypothesis from continuity to measurability.

The central result states that any measurable, translation and $\mathrm{SO}(n)$-invariant valuation $\varphi : \mathcal{P}^n \to \mathbb{R}$ is a linear combination of the intrinsic volumes $V_0, \dots, V_n$. Since a monotone translation invariant valuation on polytopes is automatically continuous (hence measurable) by McMullen's theorem, this subsumes the continuous case. The result also covers the corresponding characterization of simple rigid motion invariant valuations: such a valuation must be a scalar multiple of the volume $V_n$. This is the polytopal analogue of Klain's characterization of volume, which underpins Alesker's proof of the Irreducibility Theorem.

A second main theorem extends Schneider's 1971 characterization of the Steiner point map to the polytopal setting: any measurable, translation and $\mathrm{SO}(n)$-equivariant, Minkowski additive map $\varphi : \mathcal{P}^n \to \mathbb{R}^n$ coincides with the Steiner point $s(P) = n\int_{S^{n-1}} v\, h_P(v)\, d\sigma(v)$.

Method: averaging a simplex representation

Hadwiger's original proof of his classification relies on a characterization of mean width that requires the valuation to be defined on all convex bodies: a 1-homogeneous translation invariant continuous valuation on $\mathcal{K}^n$ is Minkowski additive, so averaging a convex body over rotations yields a ball, and the valuation is determined by its values on balls. This step fails on polytopes because the averaging limit leaves the domain. The paper's key idea is to average a representation of the valuation rather than the bodies themselves.

The technical core is a refined version of Hadwiger's 1952 representation of weakly continuous valuations. For a weakly continuous, translation invariant, simple, 1-homogeneous valuation $\varphi : \mathcal{P}^n \to E$, the paper constructs a function $\mathrm{SO}(n)$0 on the Stiefel manifold $\mathrm{SO}(n)$1 by evaluating $\mathrm{SO}(n)$2 on an explicit family of orthogonal simplices $\mathrm{SO}(n)$3, and shows

$\mathrm{SO}(n)$4

where $\mathrm{SO}(n)$5 is the finite set of $\mathrm{SO}(n)$6-tight frames and $\mathrm{SO}(n)$7 is the face of $\mathrm{SO}(n)$8 exposed by $\mathrm{SO}(n)$9. The proof proceeds by induction on dimension, using the fact that simple 1-homogeneous valuations vanish on cylinders, together with a decomposition of arbitrary polytopes into skew cylinders.

Because $\varphi : \mathcal{P}^n \to \mathbb{R}$0 is obtained by evaluating $\varphi : \mathcal{P}^n \to \mathbb{R}$1 at continuously varying simplices, it inherits regularity: if $\varphi : \mathcal{P}^n \to \mathbb{R}$2 is measurable, so is $\varphi : \mathcal{P}^n \to \mathbb{R}$3. This is the point where measurability, rather than mere algebraic structure, enters decisively.

The regularity mechanism and equivariant vanishing

The central technical contribution is an equivariant regularity result: if $\varphi : \mathcal{P}^n \to \mathbb{R}$4 is a finite dimensional representation of $\varphi : \mathcal{P}^n \to \mathbb{R}$5 and $\varphi : \mathcal{P}^n \to \mathbb{R}$6 is measurable, translation invariant, simple, 1-homogeneous, and $\varphi : \mathcal{P}^n \to \mathbb{R}$7-equivariant, then the representation function $\varphi : \mathcal{P}^n \to \mathbb{R}$8 can be chosen continuous and $\varphi : \mathcal{P}^n \to \mathbb{R}$9-invariant, lying in $V_0, \dots, V_n$0. The proof first uses equivariance and a measure-theoretic argument (choosing rotations keeping the finitely many tight frames of a simplex inside a large-measure boundedness set) to show $V_0, \dots, V_n$1 is bounded, then averages $V_0, \dots, V_n$2 against the Haar measure using the representation $V_0, \dots, V_n$3, obtaining a continuous invariant odd function.

For $V_0, \dots, V_n$4 with trivial action, an invariant odd function on $V_0, \dots, V_n$5 vanishes identically for $V_0, \dots, V_n$6, which immediately yields the vanishing of all simple, rigid motion invariant, measurable 1-homogeneous valuations. The result extends to all degrees $V_0, \dots, V_n$7: for $V_0, \dots, V_n$8, the polarization of $V_0, \dots, V_n$9 is used to show $V_n$0 vanishes on proper orthogonal Minkowski sums, and the Canonical Simplex Decomposition then forces $V_n$1 on every orthogonal simplex, which is only consistent with $V_n$2 since $V_n$3. Combined with a lemma (due to Chen) that every polytope differs from a signed combination of orthogonal simplices by lower-dimensional terms, this gives the characterization of volume, and an induction on dimension yields the full Hadwiger theorem on polytopes.

The Steiner point theorem follows from the same machinery applied to the standard representation: for $V_n$4, $V_n$5 is irreducible, and the Canonical Decomposition of $V_n$6 shows that no nontrivial element defines an odd function, so simple equivariant 1-homogeneous valuations vanish. The two-dimensional case requires a separate argument via the centeredness of the surface area measure. The proof then shows any equivariant 1-homogeneous valuation is simple by an induction using Hadwiger's polytopal theorem on lower-dimensional restrictions, and concludes that $V_n$7 vanishes identically.

Supporting results on measurability and weak continuity

The paper also establishes infrastructure that may be of independent interest. It shows that measurable or locally bounded translation invariant valuations on $V_n$8 are automatically weakly continuous: the homogeneous components in McMullen's decomposition are real-homogeneous (via the classical fact that measurable or locally bounded solutions of Cauchy's functional equation are linear), so the Minkowski polynomial is an $V_n$9-polynomial, which by McMullen's theorem implies weak continuity. Along the way, the paper clarifies a subtlety in the literature: joint and separate continuity in the definition of weak continuity are equivalent for translation invariant valuations, a fact the author notes was implicitly relied upon in McMullen's argument but apparently not recorded explicitly.

Limitations and open questions

The paper is explicit that its result does not settle McMullen–Schneider Problem 15.4 in full. In particular, the corresponding classification for locally bounded or non-negative valuations on $\mathrm{SO}(n)$0 remains open, although the author notes the techniques apply to locally bounded valuations in several auxiliary steps. A second open direction concerns Alesker-type finiteness results for polytopes: for measurable valuations invariant under other transitive compact subgroups of $\mathrm{SO}(n)$1, the analogue of the main theorem fails, since weighted sums over faces of a fixed dimension give ample counterexamples. The author poses the question whether finiteness holds under stronger regularity assumptions such as local boundedness or continuity on $\mathrm{SO}(n)$2, and observes that constructing a valuation continuous on $\mathrm{SO}(n)$3 that does not extend to $\mathrm{SO}(n)$4 appears to be nontrivial. The paper also notes, without elaboration, that Schneider's characterization of intermediate area measures can be transferred to polytopes by the same reduction, leaving the details to the reader.

Conclusion

The paper settles the polytopal version of Hadwiger's theorem under measurability, replacing Hadwiger's averaging of convex bodies with an averaging of an explicit Stiefel-manifold representation of the valuation, and derives the polytopal Steiner point characterization from the same equivariant regularity result. The main open problems left are the locally bounded and non-negative cases of the polytopal Hadwiger problem and the status of Alesker-type finiteness under stronger regularity assumptions.