Kakeya Needle Problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $K\subseteq\mathbb R^2$ be a bounded Lebesgue-measurable region. A closed full rotation of a unit needle within $K$ is described by continuous functions $a:[0,1]\to\mathbb R^2$ and $\theta:[0,1]\to\mathbb R$ satisfying \[ a(1)=a(0),\qquad \theta(1)=\theta(0)+2\pi,\qquad \{a(t)+u(\cos\theta(t),\sin\theta(t)):0\le u\le1\} \subseteq K\quad (0\le t\le1). \] Translations during rotation are allowed; neither convexity nor simple connectedness is imposed. Let $\mathcal K$ be the class of such regions and write $|K|$ for planar Lebesgue area.

Statement recorded in the supplied source.

Determine \[ \inf_{K\in\mathcal K}|K| \] and whether this infimum is attained. This asks for the least area needed for an actual continuous $360^\circ$ rotation returning the needle to its starting position, rather than merely a set containing a unit segment in each direction. [S2] [S3]

Short English statement

How little planar area can permit a unit needle to turn continuously through a full rotation and return to its starting position? Is there a region of minimum area?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1124 (GEO-004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hee-Kap Ahn, Sang Won Bae, Otfried Cheong, Joachim Gudmundsson, Takeshi Tokuyama and Antoine Vigneron, A Generalization of the Convex Kakeya Problem, arXiv:1209.2171, Section 1, discussion of the unrestricted problem on pages 1--2. https://arxiv.org/abs/1209.2171.
• S3 Terence Tao, Rotating needles in space: the road to the Kakeya conjecture, and why it matters, arXiv:2608.22209 (2026), Section 1, Problem 1.1 and footnotes 1--2. https://arxiv.org/abs/2608.22209.