Research Problems in Function Theory — Problem 7.22

Definitions and mathematical statement

A Jordan curve in ${\mathbb{R}}^3$ is the image of an injective continuous map from $S^1$. For disjoint Jordan curves $A,B$, define \[ \operatorname{dist}(A,B)=\inf\{\|a-b\|:a\in A,\ b\in B\}. \] Let $\operatorname{len}(A)$ be its arc length, interpreted as $\infty$ when it is nonrectifiable. The assertion attributed to Gehring is \[ \forall A,B\text{ disjoint linked Jordan curves in }{\mathbb{R}}^3, \qquad\operatorname{dist}(A,B)=1\ \Longrightarrow\ \operatorname{len}(A)\ge2\pi. \] Only the length of $A$ is bounded; no length or curvature constraint on $B$ is supplied. The source credits Gehring with the corresponding inequality using some positive absolute constant in place of $2\pi$.

The meaning of "linked" is not defined in the excerpt. Nonzero linking number and nonsplitting (no separating embedded sphere) are different notions; the latter also includes some links of linking number zero. No choice between these is silently imposed.

The structured registry status is partially solved with classification PARTIAL-PROGRESS, whereas its research summary asks to mark SOLVED-IN-LITERATURE with moderate confidence. That summary asserts the sharp $2\pi$ bound but gives no identifiable theorem or citation, and its "up to the constant" wording does not state a precise exact-constant result. Accordingly the exact $2\pi$ target is retained without promoting this vague resolution claim to a verified theorem.

Short English statement

Two linked disjoint closed curves at mutual distance one should each have length at least $2\pi$. The linking convention and the supplied inconsistent, uncited resolution assessment require review.

Sources

• Local UnsolvedMath registry, permanent ID $2307022$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 7.22."
• Hayman, Research Problems in Function Theory (2018), Problem 7.22, attributed to F. W. Gehring; https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29. The Gehring positive-constant result has no complete citation here.
• Supplied research summary refers vaguely to work on the length of linked curves, without an authoritative paper, author, theorem number, or exact-constant statement supporting its moderate-confidence solved label.