Shape containing unit line segments in all directions
Needle shown rotating inside a deltoid. At every stage of its rotation (except when an endpoint is at a cusp of the deltoid), the needle is in contact with the deltoid at three points: two endpoints (blue) and one tangent point (black). The needle's midpoint (red) describes a circle with diameter equal to half the length of the needle.
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.
In mathematics, a Kakeyaset, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance...
Kakeya may refer to: Kakeya, Shimane town Sōichi Kakeya, mathematician Kakeyaset in mathematics This disambiguation page lists articles associated with...
outline Dvir's proof of the Finite Field Kakeya Conjecture using the polynomial method. Finite Field Kakeya Conjecture: Let F q {\displaystyle \mathbb...
Nikodym sets are closely related to Kakeyasets (also known as Besicovitch sets). The existence of Nikodym sets is sometimes compared with the Banach–Tarski...
lemma and the Cauchy–Binet formula Four proofs of Cayley's formula Kakeyasets in vector spaces over finite fields Bregman–Minc inequality Dinitz problem...
the dimension of Kakeyasets. Jointly with Izabella Łaba and Terence Tao he proved that the upper Minkowski dimension of Kakeyasets in 3 dimensions is...
every Jordan curve have an inscribed square? The Kakeya conjecture – do n {\displaystyle n} -dimensional sets that contain a unit line segment in every direction...
St Andrews Besicovitch On Kakeyas Problem and a similar one, Math. Zeitschrift vol.27, 1928, 312 Besicovitch On linear sets of points of fractal dimension...
Professor Burkard Polster of Monash University, to illustrate the mathematical Kakeya needle problem. It is claimed that comedian Bill Hicks used the phrase "Squeegee...
Prize in 1999, for his contributions to analysis and particularly to the Kakeya conjecture. He was an Invited Speaker at the International Congress of Mathematicians...
developed by Carleson and Per Sjölin has been standard in the study of the Kakeya problem. In 1974 he solved the extension problem for quasiconformal mappings...
to the triangle's sides. [1] A deltoid was proposed as a solution to the Kakeya needle problem. Astroid, a curve with four cusps Circular horn triangle...
restriction problem. They also found analogous results for the bilinear Kakeya problem which is based upon the X-ray transform instead of the Fourier transform...
words, the cardinality of the set of transcendentals (denoted ℶ 1 {\displaystyle \beth _{1}} ) is greater than that of the set of algebraic numbers ( ℵ 0...
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