Translation-Invariant Curvature Measures

• Translation Invariant Curvature Measures are finitely additive, translation-invariant functionals that capture localized geometric data on convex bodies.
• They possess rich algebraic structures as modules over GL(n) and O(n), with key results such as the length ≤ 2 irreducibility conjecture and the Bernig embedding linking to valuations.
• Applications include Steiner-type volumetric expansions and the classification of angular measures, which bridge convex geometry, integral geometry, and valuation theory.

Translation-invariant curvature measures are finitely additive, translation-invariant, and local-valued functionals on the space of convex bodies in $\mathbb{R}^n$, encoding localized geometric data such as Federer's curvature measures and derived from the intersection theory of convex geometry, integral geometry, and valuation theory. These measures are equipped with rich algebraic and representation-theoretic structures, particularly as modules over general linear and orthogonal groups, and are foundational for understanding local geometric invariants on convex bodies, polytopes, and submanifolds.

1. Definitions and Structural Properties

A convex body $K \subset \mathbb{R}^n$ is a non-empty compact convex subset. A translation-invariant curvature measure is a map

$$\Phi: \mathbb{K}(\mathbb{R}^n) \to \mathcal{M}(\mathbb{R}^n)$$

assigning to each convex body $K$ a signed Borel measure $\Phi(K, \cdot)$, satisfying three essential properties:

• Locality: $\Phi(K,\beta) = \Phi(L,\beta)$ if $K \cap U = L \cap U$ for some open $U \supseteq \beta$.
• Translation-invariance: $\Phi(K+x, \beta + x) = \Phi(K,\beta)$ for all $x\in\mathbb{R}^n$.
• Continuity: Weak continuity in the Hausdorff topology: $K_i \to K$ implies $\Phi(K_i) \Rightarrow \Phi(K)$.

The collection $\mathrm{Curv}(\mathbb{R}^n)$ of all such measures forms a real (or complex) vector space, naturally graded by homogeneity (degree $r$) and parity (even/odd under reflection): $$\Phi = \Phi_0 + \Phi_1 + \cdots + \Phi_n,\qquad \mathrm{Curv}_r = \mathrm{Curv}_r^+ \oplus \mathrm{Curv}_r^-.$$ Prime examples include Federer's curvature measures $C_r$, which are obtained by restricting the $r$th support measure $\Theta_r$ to normal cycles, and the Lebesgue measure localization $K \mapsto \mathrm{Vol}(K\cap\cdot)$.

For $r=0$ and $r=n$, the structure is especially simple: $C_0$ globalizes to the Euler characteristic; $C_n$ is Lebesgue measure. For $r=n-1$, curvature measures are parametrized by continuous functions $f\colon S^{n-1}\to\mathbb{R}$ (modulo linear), encoding the classical Minkowski-type functionals (Schuhmacher et al., 20 Jan 2026).

2. Representation Theory and the Length-$\le2$ Conjecture

The space $\mathrm{Curv}(\mathbb{R}^n)$ admits a natural action by $\mathrm{GL}(n)$,

$$(g\cdot\Phi)(K,\beta) = \Phi(g^{-1}K, g^{-1}\beta),$$

with the subspace of total-mass-zero measures,

$$Z_r^\pm = \{\Phi \in \mathrm{Curv}_r^\pm : \Phi(K, \mathbb{R}^n) = 0 \; \forall K\},$$

being $\mathrm{GL}(n)$-invariant.

Conjecture (Length $\le 2$): For each $0\leq r\leq n-2$, $\mathrm{GL}(n)$ acts irreducibly on $Z_r^\pm$. Thus, every $\mathrm{Curv}_r^\pm$ has composition series of length at most 2 as $\mathrm{GL}(n)$-module (Schuhmacher et al., 20 Jan 2026).

This conjecture, when coupled with Alesker's irreducibility of translation-invariant valuations, implies a uniform module structure, fundamentally constraining the types of geometric phenomena translation-invariant curvature measures can encode.

3. Proved Cases and Main Technical Tools

For $r=0$ and $r=n-2$, the conjecture is verified:

• For $r=0$: The only nontrivial measure is the Euler-kernel, giving a $1$-dimensional irreducible module.
• For $r=n-2$: Injective Bernig embedding $$B: Z_{n-2} \to \mathrm{Val}_{n-1}(\mathbb{R}^n, \mathbb{R}^n)$$ relates $Z_{n-2}$ with vector-valued valuations. Highest-weight theory and precise comparison of $\mathrm{K}$-types via (g,K)-module methods establish irreducibility.

The proof methods deploy:

• Kiderlen–Weil decomposition: Decomposition of measures on polytopes through facewise sums involving cone-valuations;
• Weil's valuation property: The valuation identity holds for all such measures;
• Globalization map: $$\mathrm{glob}:\Phi \mapsto [K\mapsto\Phi(K,\mathbb{R}^n)]$$ maps curvature measures onto valuations, with kernel $Z_r$;
• Bernig embedding: $$Z_r\hookrightarrow \mathrm{Val}_{r+1}(\mathbb{R}^n, \mathbb{R}^n)$$, injective and $\mathrm{GL}(n)$-equivariant;
• (g,K)-module and Casselman–Wallach theory: For precise multiplicity and length bounds.

These approaches reduce the structural problem for curvature measures to explicit and tractable questions in representation theory (Schuhmacher et al., 20 Jan 2026).

4. Classification of Angular Curvature Measures

Angular curvature measures are characterized by the property that their local densities on any polytope are supported on faces and proportional to the corresponding external angles. Any such measure of degree $k < n-1$ is determined by an even, 2-homogeneous polynomial on the Plücker image of $\mathrm{Gr}_k(\mathbb{R}^n)$, and the correspondence

$$f \longmapsto \Phi_f, \qquad \Phi_f(P,U) = \sum_{\dim F = k} f(\mathrm{span}F)\gamma(F,P)\mathrm{Vol}_k(F\cap U),$$

yields a canonical basis. For $k=n-1$, angular curvature measures correspond to even smooth functions on $S^{n-1}$, with basis described by spherical harmonics (Wannerer, 2018).

Explicitly, for $k<n-1$,

$$\dim\,\mathrm{Curv}_k^{\mathrm{ang}}(\mathbb{R}^n) = \frac{1}{n-k+1}\binom{n}{k}\binom{n+1}{k+1}.$$

This structure is stable under $\mathrm{O}(n)$-action, giving rise to irreducible modules classified by highest weight $2\omega_{n-k}$.

5. Smooth, Invariant, and Covariant Curvature Measures

The full theory of smooth translation-invariant curvature measures involves integration of specific primitive differential forms over the normal cycle, with the global structure described in three components:

• Representation Correspondence: Every smooth, translation-invariant, $\mathrm{SO}(n)$-covariant, $k$-homogeneous curvature measure with values in an irreducible module $\Gamma_{[\lambda]}$ is in bijection with a unique primitive form of bidegree $(k, n-1-k)$ on the sphere bundle.
• Model and Canonical Forms: Construction of such forms uses model forms in $(x,y)$ coordinates, with symmetrization to irreducibles via Young symmetrizers and trace projections. The resulting spaces admit explicit decomposition through the Lefschetz decomposition and symplectic structure on the contact manifold $S\mathbb{R}^n$.
• Globalization and Kernel: The globalization map from curvature measures to valuations admits explicit characterization of its kernel in terms of three classes of relations among forms, and its image provides a countable basis of translation-invariant valuations in each degree (Saienko, 2019).

This refined perspective demonstrates that curvature measures are localized versions of valuations, providing more granular control over local geometric content.

6. Characterization of Federer's Curvature Measures and Axiomatic Results

Schuhmacher–Wannerer's structure theorem provides an axiomatic characterization: any $\mathrm{SO}(n)$-invariant translation-invariant curvature measure is necessarily a linear combination of Federer's curvature measures $C_r$. Notably, this result holds without any nonnegativity assumption and follows from Hadwiger's theorem on globalizations combined with the vanishing of $SO(n)$-invariants in the kernel $Z$. This strengthens earlier results by Schneider, which required a positivity assumption (Schuhmacher et al., 20 Jan 2026).

Thus, Federer's measures are uniquely determined among all $\mathrm{SO}(n)$-invariant translation-invariant curvature measures by the basic invariance and locality axioms.

7. Steiner-Type Expansions and Local Formulas

Classical Steiner-type formulas connect curvature measures to volumetric expansions such as

$$\mathrm{Vol}(K \oplus tB^n \cap \beta) = \sum_{i=0}^n t^{n-i} \frac{\Theta_i(K, \beta \times S^{n-1})}{\omega_{n-i}},$$

with support measures $\Theta_r$, and yield explicit localizations for intrinsic volumes. The intrinsic volumes $V_i$ are recovered as globalizations of the kernels $C_i$ in the corresponding component of $\mathrm{Curv}_i$. The normal cycle formalism admits explicit invariant differential form representations for curvature measures, unifying geometric, combinatorial, and measure-theoretic viewpoints (Schuhmacher et al., 20 Jan 2026).

Table: Fundamental Structural Aspects

Aspect	Description	Source (arXiv ID)
Grading	Degree ($r$) and parity (even/odd under reflection)	(Schuhmacher et al., 20 Jan 2026)
GL(n)-action	Natural, irreducibility in kernel conjectured (length $\le 2$), verified for $r=0$, $n-2$	(Schuhmacher et al., 20 Jan 2026)
Axioms	Locality, translation-invariance, continuity; valuation property holds	(Schuhmacher et al., 20 Jan 2026)
Angularity	Classifies via 2-homogeneous polynomials; links to external angles and Plücker coordinates	(Wannerer, 2018)
Smooth theory	Differential forms on the sphere bundle, Lefschetz decomposition, explicit basis via globalisation	(Saienko, 2019)

The study of translation-invariant curvature measures thus bridges geometric measure theory, convex geometry, and modern representation theory, providing a comprehensive local theory underpinning both global valuation invariants and finer angular/geometric analysis.