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. Besicovitch showed that there are Besicovitch sets of measure zero. 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. Besicovitch showed that there are Besicovitch sets of measure zero. A Kakeya needle set (sometimes also known as a Kakeya set) is a (Besicovitch) 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. The Kakeya needle problem asks whether there is a minimum area of a region 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 Pál showed. Kakeya seems to have suggested that the Kakeya set 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. Besicovitch was able to show that there is no lower bound > 0 for the area of such a region D, in which a needle of unit length can be turned round. 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 showing such a set could have arbitrarily small measure was from 1919. The problem may have been considered by analysts before that. One method of constructing a Besicovitch set (see figure for corresponding illustrations) is known as a 'Perron tree' after Oskar Perron who was able to simplify Besicovitch's original construction: take a triangle with height 1, divide it in two, and translate both pieces over each other so that their bases overlap on some small interval. Then this new figure will have a reduced total area.