Associativity in Operations on Convex Bodies

Definitions and mathematical statement

For a specified real vector space $V$ with a specified topology, let $\mathcal C$ be a specified class of compact convex subsets of $V$. A binary operation on $\mathcal C$ is a map \[ \star:\mathcal C\times\mathcal C\longrightarrow\mathcal C. \] It is associative if \[ (K\star L)\star M=K\star(L\star M) \qquad\text{for every }K,L,M\in\mathcal C. \] These definitions provide a conditional framework; the supplied record does not specify $V$, its dimension or topology, or the precise class $\mathcal C$.

Source question. What is the role of associativity in classifying operations between arbitrary compact convex sets?

A precise classification problem would require specifying the admissible operations and the intended classification criterion, and then determining which admissible operations satisfy the displayed identity and how imposing it changes the classification. The source supplies neither these specifications nor any additional axioms for the operations. It also does not state whether the empty set is allowed, whether lower-dimensional sets are included, or whether an origin-related restriction is intended. Consequently, the question is retained as a research question with unresolved parameters; it does not determine a unique classification theorem or conjecture.

Short English statement

The problem asks how requiring associativity affects classifications of operations on compact convex sets. Associativity means that combining three sets gives the same result under either placement of parentheses. The source leaves the ambient setting, admissible sets, operation axioms, and classification target unspecified, so a unique precise conjecture cannot be extracted. Its open status is supplied curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30003004$, source problem code catalogue entry, title Associativity in Operations on Convex Bodies, in the Oberwolfach Reports Open Problems collection. The supplied record labels the statement exact and reports that it was retained byte-for-byte; this is a supplied textual verification, not mathematical review. No individual proposer is identified.
• Supplied report citation: Convex Geometry and its Applications (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/56, pp. 3179--3269. https://doi.org/10.4171/owr/2015/56. The registry gives proposed year 2015, while the supplied report title uses 2016; this discrepancy is unresolved. The background attributes the question to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source: Operations between convex sets. https://doi.org/10.4171/JEMS/422.
• Supplied literature source: Classification of convex-set operations. https://arxiv.org/abs/1205.4327.
• Supplied literature source: Operations between functions. https://intlpress.com/site/pub/files/_fulltext/journals/cag/2018/0026/0004/CAG-2018-0026-0004-a005.pdf.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that partial classifications exist while the role of associativity in the full arbitrary-operation classification remains explicitly unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The background expressly calls for independent verification of time-sensitive status and cited claims.