Skip to content
ProvedAnalysis·1971

Kakeya Conjecture (2D)

A planar set can contain a unit segment in every direction and have area zero, but its Hausdorff dimension is still 2.

Formula

Kakeya conjecture (2D, proved)
K⊂R2,  K contains a unit segment in every direction⟹dim⁡HK=2K\subset\mathbb{R}^2,\; K\text{ contains a unit segment in every direction}\quad\Longrightarrow\quad \dim_H K=2

A Besicovitch set in the plane has full Hausdorff dimension — proved by Davies in 1971.

Besicovitch construction
∀ ε>0,  ∃ K⊂R2:  ∣K∣<ε,  K contains a unit segment in every direction\forall\,\varepsilon>0,\;\exists\, K\subset\mathbb{R}^2:\;|K|<\varepsilon,\; K\text{ contains a unit segment in every direction}

Besicovitch (1919) showed Kakeya sets can have arbitrarily small Lebesgue measure, despite containing all directions.

Kakeya maximal conjecture
fδ∗(e)=sup⁡a1∣Teδ(a)∣∫Teδ(a)∣f(x)∣ dx,∥fδ∗∥Ln(Sn−1)≤Cεδ−ε∥f∥Ln(Rn)f_\delta^*(e)=\sup_a \frac{1}{|T_e^\delta(a)|}\int_{T_e^\delta(a)} |f(x)|\,dx,\quad \|f_\delta^*\|_{L^n(S^{n-1})}\le C_\varepsilon\delta^{-\varepsilon}\|f\|_{L^n(\mathbb{R}^n)}

A standard normalized form of the directional maximal estimate; it controls directions on the sphere and implies the Kakeya dimension conjecture.

Summary

The planar Kakeya problem is the cleanest entry point to the whole Kakeya family. Besicovitch showed that a needle can point in every direction inside sets of arbitrarily small area, even measure zero. Davies proved the two-dimensional Kakeya dimension statement: every planar Besicovitch set still has full Hausdorff dimension 2.

Sources

Some remarks on the Kakeya problem

1971

On Kakeya's problem and a similar one

1928

New Proof Threads the Needle on a Sticky Geometry Problem

2023

Videos

Thumbnail for Kakeya's Needle Problem - Numberphile

Kakeya's Needle Problem - Numberphile

Numberphile
Thumbnail for The Kakeya needle problem (the squeegee approach)

The Kakeya needle problem (the squeegee approach)

Mathologer