 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. besicovitch showed that there are kakeya needle sets of arbitrarily small positive measure. ==kakeya needle problem== 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 360Â°. this question was first posed, for convex regions, by . the minimum area for convex sets is achieved by an equilateral triangle of height&nbsp;1 and area 1/&radic;3, as pÃ¡l 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 sets== thumb|a "sprouting" method for constructing a kakeya set of small measure. shown here are two possible ways of dividing our triangle and overlapping the pieces to get a smaller set, the first if we just use two triangles, and the second if we use eight. notice how small the sizes of the final figures are in comparison to the original starting figure. besicovitch{{cite journal | last=besicovitch | first=abram | authorlink=abram samoilovitch besi
Discrete_geometry Harmonic_analysis Real_analysis