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 Besicovitch set. A Kakeya needle set (sometimes also known as a Kakeya set) is a set in the plane with a stronger property, that a unit line segment can be rotated continuously through 360 degrees within it, returning to its original position. Again, the disk of radius 1/2 is an example of a Kakeya needle set. Much of the research in this area has studied the problem of how small such sets can be, first asked by Sōichi Kakeya in 1917. Abram Besicovitch proved in 1920 that there are Besicovitch sets in the plane of measure zero and in 1928 that there are Kakeya needle sets in the plane of arbitrarily small positive measure. There are no Kakeya needle sets of measure 0. The Kakeya conjecture states that Besicovitch sets in n-dimensional space must have Hausdorff dimension n; it remains open for n>3. These questions belong to geometric measure theory.

Besicovitch sets of measure zero In 1920, while considering a problem of integration, Besicovitch showed that there are Besicovitch sets in the plane of measure zero. Such sets can be constructed using the Perron tree method explained below, without the need for Pál joins. There are also other methods; for example, Kahane uses Cantor sets to construct a Besicovitch set of measure zero in the plane.

Kakeya needle problem The Kakeya needle problem asks whether there is a minimum area of a region D {\displaystyle D} in the plane, in which a needle of unit length can be turned through 360°. This question was first posed, for convex regions, by Sōichi Kakeya (1917). The minimum area for convex sets is achieved by an equilateral triangle of height 1 and area 1/√3, as Gyula Pál showed in 1920. Kakeya seems to have suggested that the Kakeya set D {\displaystyle D} 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.

Kakeya needle sets of arbitrarily small measure

Abram Besicovitch was able to show in 1928 that there is no lower bound > 0 for the area of such a region D {\displaystyle D} , in which a needle of unit length can be turned around. That is, for every ε > 0 {\displaystyle \varepsilon >0} , there is region of area ε {\displaystyle \varepsilon } within which the needle can move through a continuous motion that rotates it a full 360 degrees. The AMS produced a film in 1962 in which Besicovitch explains a construction of such a set. One method of constructing a Kakeya needle set of arbitrary small area (and also of a Besicovitch set of measure zero) is known as a "Perron tree", named after Oskar Perron who simplified 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 r {\displaystyle r} 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 circle segments at the corners of the "N". The swept area is proportional to the angle which is proportional to 1 / r {\displaystyle 1/r} , and thus the swept out area can be made arbitrarily small by choosing an appropriately large r {\displaystyle r} . The construction starts with any triangle with height 1 and some substantial angle at the top through which the needle can easily sweep. (See figure to the right.) 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 (arbitrarily small) 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.

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