Lower Bounds for Convex Decompositions of Crossing Curves

Definitions and mathematical statement

Fix a dimension $d$ in the intended range, which the supplied record does not specify. Let $\mathcal C_d$ denote the intended class of $(d+1)$-crossing curves in ${\mathbb{R}}^d$. For $\gamma\in\mathcal C_d$ and an integer $m\geq1$, let $\mathcal D_m(\gamma)$ denote the set of admissible decompositions of $\gamma$ into exactly $m$ convex curves. These are placeholders for geometric notions whose definitions are absent from the supplied record; they do not supply missing hypotheses.

Provided the following set is nonempty, define \[ M(d)=\min\left\{m\in{\mathbb{Z}}_{\geq1}:\ \forall\gamma\in\mathcal C_d,\quad \mathcal D_m(\gamma)\ne\varnothing\right\}. \] The source asks for nontrivial lower bounds on $M(d)$. More explicitly, seek an integer-valued bound $B(d)\geq1$ and establish \[ M(d)\geq B(d). \] Under the displayed definition, this is equivalent to proving \[ \forall m\in{\mathbb{Z}}_{\geq1},\quad m<B(d)\ \Longrightarrow\ \exists\gamma\in\mathcal C_d\text{ such that }\mathcal D_m(\gamma)=\varnothing. \] The source specifies neither a target bound nor a precise meaning of "nontrivial."

The geometric formulation remains unresolved until the meanings of crossing curve, convex curve, and decomposition are fixed. In particular, the record does not specify the parameter domain, regularity or injectivity of curves, how hyperplane intersections or contacts are counted, or how endpoints and shared subdivision points are treated. The displayed definition follows the literal phrase "into $m$ convex curves." Interpreting this as "into at most $m$ convex curves" requires clarification or a justification that the two conventions agree for the intended class.

Short English statement

Determine meaningful lower bounds for the least number of convex pieces sufficient to split every $(d+1)$-crossing curve in ${\mathbb{R}}^d$. The supplied record leaves the geometric definitions and dimension range unspecified, so the formal statement is conditional on resolving those choices. Its partially solved status is supplied curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30003473$, title "Lower Bounds for Convex Decompositions of Crossing Curves," exact source problem code catalogue entry. The record attributes the problem to the Oberwolfach Reports workshop collection, gives proposed year $2017$, and names no individual proposer.
• Supplied source citation: Discrete Geometry (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/19, pp. 1135--1205. https://doi.org/10.4171/owr/2017/19. The citation is report-level provenance; no exact problem page is supplied.
• Supplied literature reference, labeled "Bárány Matoušek and Pór on crossing curves": https://arxiv.org/abs/1309.1147.
• Supplied literature reference, labeled "related geometric Ramsey construction": https://doi.org/10.1007/s00208-015-1232-x.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and reports a decomposition theorem giving $M(d)\leq4d$, together with $M(2)=4$ and $M(3)\leq22$, while describing general lower bounds and sharp values as unresolved. These claims and this status are supplied dated curation, not independently verified here. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and explains that notation, typography, and sentence structure were repaired while preserving mathematical content. This supplied editorial label does not establish the missing geometric definitions or constitute mathematical review of this draft.