Kakeya set
shape containing unit line segments in all directions
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.
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This entry incorporates text from “Kakeya set” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.