Invariant Valuations for Noncompact Groups

Definitions and mathematical statement

For the explicitly named example, let $p,q\geq 1$ be integers, put $V={\mathbb{R}}^{p+q}$, and define \[ Q_{p,q}(x)=\sum_{i=1}^{p}x_i^2-\sum_{i=p+1}^{p+q}x_i^2, \qquad O(p,q)=\{g\in\operatorname{GL}(V):Q_{p,q}(gx)=Q_{p,q}(x)\text{ for all }x\in V\}. \] This gives the indefinite orthogonal example mentioned in the source; the source does not fix $p,q$, a signature range, or a dimension.

For terminology in this linear model, let $\mathcal K(V)$ be the collection of compact convex subsets of $V$, including the empty set. For an abelian group $A$, an ordinary $A$-valued valuation is a map $\phi:\mathcal K(V)\to A$ satisfying $\phi(\varnothing)=0$ and \[ \phi(K\cup L)+\phi(K\cap L)=\phi(K)+\phi(L) \] whenever $K,L\in\mathcal K(V)$ and $K\cup L$ is convex. If a group $G$ acts on $\mathcal K(V)$, such a valuation is $G$-invariant when \[ \phi(gK)=\phi(K)\qquad(g\in G,\ K\in\mathcal K(V)). \] These definitions supply ordinary valuation terminology only. The source does not specify the value space, continuity, translation invariance, homogeneity, or a meaning of generalized valuation; generalized valuations need not be pointwise functions on all convex bodies.

Research question as supplied. For noncompact transformation groups $G$, with $O(p,q)$ as an explicit example, develop an appropriate generalized invariant valuation framework and corresponding kinematic formulas extending or replacing those of compact-group invariant valuation theory.

The supplied record does not determine a single quantified conjecture or a particular kinematic identity. It leaves unspecified the admissible groups and actions, ambient spaces, valuation spaces, regularity assumptions, averaging measures, and convergence or distributional interpretation of the proposed formulas. In particular, it does not state whether translations are included in the acting group, or whether the question is intended for every noncompact group or for selected classes. These choices remain unresolved and must be supplied before a precise existence or classification assertion can be formulated.

The supplied dated curation labels the question partially solved and reports classifications of continuous and generalized invariant valuations for indefinite orthogonal groups, while emphasizing that the extracted question is broader than that class. This is a supplied literature assessment, not an independently verified theorem or status determination.

Short English statement

The question asks how invariant valuations and kinematic formulas should be formulated for noncompact groups, especially indefinite orthogonal groups $O(p,q)$. It is a research-program question: the source does not specify enough hypotheses to state one precise conjecture. The supplied curation reports partial progress for indefinite orthogonal groups, subject to independent verification and human mathematical review.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30003003$, title Invariant Valuations for Noncompact Groups, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2015.
• Supplied report citation: Convex Geometry and its Applications (2016), Oberwolfach Reports, pp. 3179--3269, DOI 10.4171/owr/2015/56. Original Oberwolfach report: https://doi.org/10.4171/owr/2015/56. The record attributes its extraction to workshop open-problem discussions; it supplies no problem-specific page or named speaker.
• Supplied literature reference: Valuations for indefinite orthogonal groups, https://arxiv.org/abs/1602.08760.
• Supplied literature reference: Invariant Crofton formulas, https://arxiv.org/abs/1612.01625.
• Literature assessment date, copied exactly: 2026-08-21. This assessment is supplied dated curation, not independently verified. It labels the problem partially_solved and reports classifications of continuous and generalized invariant valuations for indefinite orthogonal groups, while qualifying that the broader noncompact-group question extends beyond that class. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that notation and publication prose were repaired without changing its substance. This is supplied editorial metadata, not independent verification of the report, its attribution, or mathematical correctness.