Nonpointwise, Nonassociative Operations on Convex Objects

Definitions and mathematical statement

A subset $K$ of a real vector space $V$ is convex if $(1-t)x+ty\in K$ for all $x,y\in K$ and $t\in[0,1]$. A real-valued function $f:D\to{\mathbb{R}}$ on a convex domain $D\subseteq V$ is convex if \[ f((1-t)x+ty)\leq (1-t)f(x)+tf(y) \qquad(x,y\in D,\ t\in[0,1]). \] These definitions provide terminology; the source does not specify an ambient space, a domain, or whether functions must be real-valued rather than extended-real-valued.

If binary operations on a class $\mathcal X$ are intended, such an operation is a map $\star:\mathcal X\times\mathcal X\to\mathcal X$. It is associative if \[ (A\star B)\star C=A\star(B\star C) \qquad\text{for all }A,B,C\in\mathcal X. \] For a class $\mathcal F$ of real-valued functions on a common domain $D$, one possible meaning of a pointwise operation is that there exists a single map $\Phi:{\mathbb{R}}^2\to{\mathbb{R}}$ such that \[ (f\star g)(x)=\Phi(f(x),g(x)) \qquad\text{for all }f,g\in\mathcal F\text{ and }x\in D. \] This is a conditional interpretation, not a definition supplied by the record. In particular, the record does not define pointwise operations on convex sets or specify a representation through which that term should be interpreted.

Source problem. Find classification results for operations on convex functions or convex sets that are neither necessarily pointwise nor associative.

Unresolved scope. The wording is retained as a research request. It does not determine a single precise classification theorem: the classes of convex objects, the arity and codomain of the operations, the remaining structural hypotheses, and the equivalence relation or normal forms for classification are unspecified. The phrase "neither necessarily pointwise nor associative" may mean that neither property is assumed; it does not unambiguously require every operation under consideration to fail both properties. No such additional requirements are imposed here. A precise theorem formulation requires these choices to be supplied.

Short English statement

The problem asks for classification results about operations on convex functions or convex sets without necessarily assuming pointwise behavior or associativity. The source leaves the object classes, admissible operations, other hypotheses, and meaning of classification unspecified, so this is a research direction requiring further formulation. The supplied record labels it open; that status and its literature assessment have not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003005$, source problem code catalogue entry, title Nonpointwise, Nonassociative Operations on Convex Objects, in the Oberwolfach Reports Open Problems collection. The record marks the statement as exact and says it was retained byte-for-byte; this describes supplied textual verification, not mathematical completeness. No individual proposer is identified.
• Supplied report citation: Convex Geometry and its Applications (2016), Oberwolfach Reports, pp. 3179--3269, DOI 10.4171/owr/2015/56: https://doi.org/10.4171/owr/2015/56. The supplied background attributes the problem collection to workshop open-problem discussions. The record gives proposed year 2015; the report title year and DOI are retained as supplied.
• Supplied literature reference: Operations between convex sets, https://doi.org/10.4171/JEMS/422.
• Supplied literature reference: Classification of convex-set operations, https://arxiv.org/abs/1205.4327.
• Supplied literature reference: Operations between functions, https://intlpress.com/site/pub/files/_fulltext/journals/cag/2018/0026/0004/CAG-2018-0026-0004-a005.pdf.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and states that no general classification is known for operations that are neither pointwise nor associative, while existing theorems impose one or both hypotheses. This is supplied dated curation derived from the OWR record, not an independently verified assessment. Its time-sensitive status and claims about the cited literature require independent verification.