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Week 23 · ClassicalPartial progress
Primary source

Kakeya Conjecture — Does a Needle-Set Have Full Dimension?

Research level· Posed 1928

Problem

A Kakeya set in Rn\mathbb{R}^n contains a unit segment in every direction. Besicovitch (1928) showed such sets can have Lebesgue measure zero. The Kakeya conjecture: every Kakeya set has Minkowski and Hausdorff dimension exactly nn.

Why it matters

In the plane (n=2n=2), dimension 2 was proved by Davies (1971). For n=3n=3 the problem stayed open until Hong Wang & Joshua Zahl (2025-02-24) announced a complete resolution — a major event in analytic-combinatorics. For n≥4n \ge 4 the conjecture is still completely open.

Progress so far

n=2n=2 solved by Davies (1971). n=3n=3 resolved by Wang & Zahl on 2025-02-24 (arXiv:2502.17655). For n≥4n \ge 4 the conjecture remains open. The polynomial method (Dvir 2008 finite-field Kakeya) is a key tool but hard to extend.

Further reading

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