Convexity of Odd-Power Sum-of-Squares Cones

Definitions and mathematical statement

For a real topological vector space $V$, a subset $C\subseteq V$ is a closed convex cone if it is nonempty, closed in the specified topology of $V$, and satisfies \[ \forall f,g\in C\;\forall a,b\in{\mathbb{R}}_{\geq 0},\qquad af+bg\in C. \] This is a general definition only; the supplied record does not identify the ambient space or topology for the set below.

The source question is:

For the source-intended admissible parameters $n,m,k$, is $\Sigma_{n,m}(2k+1)$ a closed convex cone?

The supplied curation describes $\Sigma_{n,m}(2k+1)$ as an odd-power sum-of-squares-root set, but gives no membership definition. In particular, it does not specify the parameter ranges, whether the underlying objects are polynomials or homogeneous forms, the degree convention, or the ambient topology. Consequently, the notation cannot be expanded into a fully specified mathematical assertion from the supplied evidence alone.

The supplied literature assessment attributes closedness to the source and describes convexity as difficult and conjectural. Thus closedness is presented in the curation as established, while convexity remains the stated unresolved issue. These are supplied source assessments, not independently verified mathematical conclusions.

Short English statement

The question asks whether the source's odd-power sum-of-squares-root set $\Sigma_{n,m}(2k+1)$ is a closed convex cone. The supplied assessment says closedness is proved and convexity remains open. A precise formulation requires the missing definition of this set, its admissible parameters, and its ambient space and topology.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005460$, source problem code catalogue entry, title "Convexity of Odd-Power Sum-of-Squares Cones." The registry labels the problem open and the statement corrected_verified; these are supplied metadata labels. Its verification note reports direct checking of the official report and restoration of definitions, but those definitions are absent from the supplied record.
• Real Algebraic Geometry with a View toward Koopman Operator Methods (2023), Oberwolfach Reports, pp. 741--811, DOI 10.4171/owr/2023/14. The extracted question carries page number 779. Supplied source URL and literature-source link, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2023/14. No individual proposer is identified in the supplied record.
• Literature assessment dated exactly 2026-08-22: the supplied curation says the source proves closedness, leaves convexity difficult and conjectural, and that no primary result found in that review establishes or refutes convexity for the stated general set. This is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.