Kakeya set

In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensional space, forms a Kakeya set. Much of the research in this area has studied the problem of how small such sets can be. Abram Besicovitch showed that there are Besicovitch sets of measure zero.
A Kakeya needle set is a set in the plane with a stronger property, that a unit line segment can be rotated continuously through 180 degrees within it, returning to its original position with reversed orientation. Again, the disk of radius 1/2 is an example of a Kakeya needle set.

Kakeya needle problem

The Kakeya needle problem asks whether there is a minimum area of a region in the plane, in which a needle of unit length can be turned through 360°. This question was first posed, for convex regions, by. The minimum area for convex sets is achieved by an equilateral triangle of height 1 and area 1/, as Pál showed.
Kakeya seems to have suggested that the Kakeya set of minimum area, without the convexity restriction, would be a three-pointed deltoid shape. However, this is false; there are smaller non-convex Kakeya sets.

Besicovitch needle sets

was able to show that there is no lower bound > 0 for the area of such a region, in which a needle of unit length can be turned around. That is, for every, there is region of area within which the needle can move through a continuous motion that rotates it a full 360 degrees. This built on earlier work of his, on plane sets which contain a unit segment in each orientation. Such a set is now called a Besicovitch set. Besicovitch's work from 1919 showed such a set could have arbitrarily small measure, although the problem may have been considered by analysts before that.
One method of constructing a Besicovitch set is known as a "Perron tree", named after Oskar Perron who was able to simplify Besicovitch's original construction. The precise construction and numerical bounds are given in Besicovitch's popularization.
The first observation to make is that the needle can move in a straight line as far as it wants without sweeping any area. This is because the needle is a zero width line segment. The second trick of Pál, known as Pál joins, describes how to move the needle between any two locations that are parallel while sweeping negligible area. The needle will follow the shape of an "N". It moves from the first location some distance up the left of the "N", sweeps out the angle to the middle diagonal, moves down the diagonal, sweeps out the second angle,
and then moves up the parallel right side of the "N" until it reaches the required second location. The only non-zero area regions swept are the two triangles of height one and the angle at the top of the "N". The swept area is proportional to this angle which is proportional to, and thus the swept out area can be made arbitrarily small by choosing an appropriately large.
The construction starts with any triangle with height 1 and some substantial angle at the top through which the needle can easily sweep. The goal is to do many operations on this triangle to make its area smaller while keeping the directions through which the needle can sweep the same. First, consider dividing the triangle into two and translating the pieces over each other so that their bases overlap in a way that minimizes the total area. The needle is able to sweep out the same directions by sweeping out those given by the first triangle, jumping over to the second, and then sweeping out the directions given by the second. The needle can jump triangles using the "N" technique because the two lines at which the original triangle was cut are parallel. In this construction, the line segment actually leaves the original overlapping triangle area and sweeps out new additional area.
Now, we divide our triangle into 2ⁿ subtriangles. The figure shows eight. For each consecutive pair of triangles, perform the same overlapping operation we described before to get half as many new shapes, each consisting of two overlapping triangles. Next, overlap consecutive pairs of these new shapes by shifting them so that their bases overlap in a way that minimizes the total area. Repeat this n times until there is only one shape. Again, the needle is able to sweep out the same directions by sweeping those out in each of the 2ⁿ subtriangles in order of their direction. The needle can jump consecutive triangles using the "N" technique because the two lines at which these triangle were cut are parallel.
What remains is to compute the area of the final shape. Due to difficulty and length constraints the final argument cannot be fully included. Instead, an example will be shown. Looking at the figure, it can be seen that the 2ⁿ subtriangles overlap a lot. All of them overlap at the bottom, half of them at the bottom of the left branch, a quarter of them at the bottom of the left left branch, and so on. Suppose that the area of each shape created with i merging operations from 2ⁱ subtriangles is bounded by A_i. Before merging two of these shapes, they have area bounded by 2A_i. Then, move the two shapes together such that that they overlap as much as possible. In the worst case, these two regions are two 1 by ε rectangles perpendicular to each other so that they overlap at an area of only ε². But the two shapes that we have constructed, if long and skinny, point in much of the same direction because they are made from consecutive groups of subtriangles. The handwaving states that they overlap by at least 1% of their area. Then the merged area would be bounded by A_i+1 = 1.99 A_i. The area of the original triangle is bounded by 1. Hence, the area of each subtriangle is bounded by A₀ = 2^-n and the final shape has area bounded by A_n = 1.99ⁿ × 2^-n. In actuality, a careful summing up of all areas that do not overlap shows that the area of the final region is much bigger, namely, 1/n. As n grows, this area shrinks to zero. A Besicovitch set can be created by combining six rotations of a Perron tree created from an equilateral triangle. A similar construction can be made with parallelograms.
There are other methods for constructing Besicovitch sets of measure zero aside from the 'sprouting' method. For example, Kahane uses Cantor sets to construct a Besicovitch set of measure zero in the two-dimensional plane.
Image:KakeyaNeedleSet3.GIF|thumb|A Kakeya needle set constructed from Perron trees.
In 1941, H. J. Van Alphen showed that there are arbitrary small Kakeya needle sets inside a circle with radius 2 + ε. Simply connected Kakeya needle sets with smaller area than the deltoid were found in 1965. Melvin Bloom and I. J. Schoenberg independently presented Kakeya needle sets with areas approaching to, the Bloom-Schoenberg number. Schoenberg conjectured that this number is the lower bound for the area of simply connected Kakeya needle sets. However, in 1971, F. Cunningham showed that, given ε > 0, there is a simply connected Kakeya needle set of area less than ε contained in a circle of radius 1.
Although there are Kakeya needle sets of arbitrarily small positive measure and Besicovitch sets of measure 0, there are no Kakeya needle sets of measure 0.

Kakeya conjecture

Statement

The same question of how small these Besicovitch sets could be was then posed in higher dimensions, giving rise to a number of conjectures known collectively as the Kakeya conjectures, and have helped initiate the field of mathematics known as geometric measure theory. In particular, if there exist Besicovitch sets of measure zero, could they also have s-dimensional Hausdorff measure zero for some dimensions less than the dimension of the space in which they lie? This question gives rise to the following conjecture:
This is known to be true for n = 1, 2 but only partial results are known in higher dimensions.
The Kakeya conjecture is closely related to the restriction conjecture, Bochner-Riesz conjecture and the local smoothing conjecture.
In February 2025, a claimed proof for the case n = 3 was posted on arXiv by Hong Wang and Joshua Zahl. The Kakeya conjecture in three dimensions is described as "one of the most sought-after open problems in geometric measure theory", and the claimed proof is considered to be a breakthrough.

Kakeya maximal function

A modern way of approaching this problem is to consider a particular type of maximal function, which we construct as follows: Denote Sⁿ⁻¹ ⊂ Rⁿ to be the unit sphere in n-dimensional space. Define to be the cylinder of length 1, radius δ > 0, centered at the point a ∈ Rⁿ, and whose long side is parallel to the direction of the unit vector e ∈ Sⁿ⁻¹. Then for a locally integrable function f, we define the Kakeya maximal function of f to be
where m denotes the n-dimensional Lebesgue measure. Notice that is defined for vectors e in the sphere Sⁿ⁻¹.
Then there is a conjecture for these functions that, if true, will imply the Kakeya set conjecture for higher dimensions:

Results

Some results toward proving the Kakeya conjecture are the following:

The Kakeya conjecture is true for n = 1 and n = 2.
In any n-dimensional space, Wolff showed that the dimension of a Kakeya set must be at least /2.
In 2002, Katz and Tao improved Wolff's bound to, which is better for n > 4.
In 2000, Katz, Łaba, and Tao proved that the Minkowski dimension of Kakeya sets in 3 dimensions is strictly greater than 5/2.
In 2000, Jean Bourgain connected the Kakeya problem to arithmetic combinatorics which involves harmonic analysis and additive number theory.
In 2017, Katz and Zahl improved the lower bound on the Hausdorff dimension of Besicovitch sets in 3 dimensions to for an absolute constant.
In 2025, Wang and Zahl posted on arXiv a potential proof of the Kakeya conjecture in the case n = 3.