Invariant Homogeneous Valuations on Convex Bodies

Definitions and mathematical statement

For a positive integer $n$, let \[ K_0^n=\{K\subset{\mathbb{R}}^n:K\text{ is compact and convex, and }0\in\operatorname{int}_{{\mathbb{R}}^n}K\}. \] Write $E=\mathbb R_+\cup\{\infty\}$, retaining the source's notation $\mathbb R_+$: the supplied record does not specify whether zero belongs to this set.

A valuation with values in $E$ is a map $\Phi:K_0^n\to E$ satisfying \[ \Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L) \] for all $K,L\in K_0^n$ for which $K\cup L$ is convex. In this case both the union and intersection belong to $K_0^n$. Addition is interpreted in the extended nonnegative range, with $a+\infty=\infty$, including $a=\infty$; the identity does not permit cancellation of infinite terms.

Let \[ \mathrm{SL}(n,{\mathbb{R}})=\{A\in{\mathbb{R}}^{n\times n}:\det A=1\},\qquad AK=\{Ax:x\in K\}. \] Invariance means \[ \Phi(AK)=\Phi(K)\qquad(A\in\mathrm{SL}(n,{\mathbb{R}}),\ K\in K_0^n). \] Under the usual interpretation of homogeneity of real degree $q$, the condition is \[ \Phi(tK)=t^q\Phi(K)\qquad(t>0,\ K\in K_0^n), \] where $tK=\{tx:x\in K\}$ and $t^q\infty=\infty$. The source specifies neither a degree nor an allowed set of degrees, and does not clarify whether the classification is to be stated separately for each degree or across all degrees.

Under the usual Hausdorff-topology interpretation, upper semicontinuity means that for every $K\in K_0^n$ and every sequence $(K_j)_{j\geq1}\subset K_0^n$, \[ d_H(K_j,K)\longrightarrow0 \quad\Longrightarrow\quad \limsup_{j\to\infty}\Phi(K_j)\leq\Phi(K), \] where $d_H$ is the Hausdorff distance induced by the Euclidean norm and the inequality uses the extended order. The supplied record does not explicitly name this topology.

Classification problem. Characterize all upper-semicontinuous, $\mathrm{SL}(n,{\mathbb{R}})$-invariant, homogeneous valuations \[ \Phi:K_0^n\longrightarrow\mathbb R_+\cup\{\infty\}. \] The value $\infty$ is expressly allowed. No finiteness or properness condition is supplied, and none is imposed here. The source does not state the admissible dimensions; the definition above describes the domain for positive integer $n$, without resolving whether the intended problem excludes any such dimensions. These dimension, degree, range, and topology qualifications require confirmation before a unique fully specified classification target can be fixed.

Short English statement

Classify homogeneous valuations on compact convex bodies whose interiors contain the origin, assuming upper semicontinuity and invariance under determinant-one linear maps. Infinite values are explicitly permitted. The intended dimensions, homogeneity degrees, convention for $\mathbb R_+$, and topology require confirmation from the source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001408$, title: Invariant Homogeneous Valuations on Convex Bodies; exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2010; no individual proposer is named.
• Mini-Workshop: Valuations and Integral Geometry (2010), Oberwolfach Reports, pp. 141--178, DOI 10.4171/owr/2010/04. Supplied source and literature reference, labelled "original Oberwolfach problem report": https://doi.org/10.4171/owr/2010/04.
• Supplied literature reference, labelled "Ludwig and Reitzner": https://annals.math.princeton.edu/wp-content/uploads/annals-v172-n2-p09-p.pdf. No title or further bibliographic details are supplied in the record.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that the report leaves the extended-range classification open, while a later real-valued classification does not settle that extension. This is supplied dated curation, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified; its accompanying verification note describes enclosing TeX notation in mathematical delimiters without changing the mathematical content. This metadata does not establish mathematical correctness or resolve the formulation qualifications above.