Monotone Cylindrical Decompositions of Compact Sets

Definitions and mathematical statement

Let $n$ be an ambient dimension, let $K\subseteq{\mathbb{R}}^n$ be compact, and let $f:K\to{\mathbb{R}}$ be a function. A partition $\mathcal P$ of $K$ is a collection of nonempty subsets of $K$ such that \[ K=\bigcup_{C\in\mathcal P}C, \qquad C\cap D=\varnothing\quad\text{for distinct }C,D\in\mathcal P. \] For $C\subseteq K$, the restriction $f|_C:C\to{\mathbb{R}}$ is given by $(f|_C)(x)=f(x)$.

The supplied question asks whether, for every such $K$ and $f$, there exists a partition $\mathcal P$ such that every $C\in\mathcal P$ is a monotone cylindrical cell and $f|_C$ is monotone. Different cells may use different coordinate orders: an order for a cell may be represented by a permutation $\sigma_C$ of $\{1,\ldots,n\}$, corresponding to the coordinates $x_{\sigma_C(1)},\ldots,x_{\sigma_C(n)}$.

Unresolved formulation qualifications. The record does not define either "monotone cylindrical cell" or monotonicity of a function on a possibly multidimensional cell. It also does not specify whether the partition must be finite, whether $K$ and $f$ must be definable in a fixed o-minimal structure, or whether $f$ must satisfy continuity or other regularity assumptions. Consequently, the preceding paragraph records the source question but does not determine a complete, unambiguous mathematical conjecture. The partition definition above imposes no cardinality condition; this is not a resolution of the missing finiteness requirement. No definability, continuity, or alternative notion of monotonicity is silently imposed.

The supplied dated assessment describes the conjecture as proved for compact sets of dimension at most two and open in arbitrary dimension, specifically including dimension three. Those claims are supplied curation, not independently verified results here, and their precise hypotheses cannot be recovered from this record.

Short English statement

The question asks whether every supplied compact set and real-valued function admit a partition into monotone cylindrical cells on which the function is monotone, allowing a different coordinate order on each cell. Precise meanings of monotonicity, the allowed partition size, and any definability or regularity hypotheses remain unresolved. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local problem registry: permanent UnsolvedMath ID $30003504$, source problem code catalogue entry, title "Monotone Cylindrical Decompositions of Compact Sets," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified. The record lists proposed year $2017$.
• Supplied report citation: O-Minimality and its Applications to Number Theory and Analysis (2018), Oberwolfach Reports, pp. 1349--1420, DOI 10.4171/owr/2017/22. https://doi.org/10.4171/owr/2017/22. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. This citation has not been independently inspected here.
• Supplied literature source, labelled "Oberwolfach Reports 2018": https://doi.org/10.4171/owr/2017/22.
• Supplied literature source, labelled "Basu et al., triangulations of monotone families": https://arxiv.org/abs/1402.0460. No fuller bibliographic attribution is supplied.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open, describes the dimension-at-most-two case as proved, and describes the arbitrary-dimensional case, including dimension three, as open. This is supplied dated curation derived from the OWR record, not independently verified literature assessment; the record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that its mathematical content was preserved while notation, typography, or omitted definitions were repaired. These are supplied registry qualifications, not mathematical verification of this draft.