Kakeya needle problem
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In mathematics, the Kakeya needle problem asks whether there is a minimum area of a region D in the plane, in which a needle can be turned through 360°. This question was first posed by Soichi Kakeya (1886-1947), a Japanese mathematician who worked mainly in mathematical analysis. In 1917 he asked this question, about convex sets.
He seems to have suggested that D of minimum area, without the convexity restriction, would be a three-pointed deltoid shape. The original problem was solved by Pal [4]. The early history of this question has been subject to some discussion, though.
Besicovitch [1] 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.
The same question was then posed in higher dimensions, giving rise to a number of conjectures known collectively as the Kakeya conjectures, and have helped initiate the field of mathematics known as geometric measure theory. A typical member of these conjectures is the following:
- Kakeya set conjecture: Define a Besicovitch set in <math>{\Bbb R}^n<math> to be a set which contains a unit line segment in every direction. Is it true that such sets necessarily have Hausdorff dimension and Minkowski dimension equal to n? This is known to be true for n=1,2 but only partial results are known in higher dimensions.
Somewhat surprisingly, these conjectures have been shown to be connected to a number of questions in other fields, notably in harmonic analysis. For instance, in 1971, Charles Fefferman [3] was able to use the Besicovitch set construction to show that Fourier series in higher dimensions, when summed spherically, do not necessarily converge in the L^p norm when <math>p \neq 2<math>.
External links
- http://www.math.ubc.ca/~ilaba/kakeya.html
- http://www.math.ubc.ca/~ilaba/comments.html
- http://www.math.ucla.edu/~tao/java/Besicovitch.html
- http://mathworld.wolfram.com/KakeyaNeedleProblem.html
References
- Abram Besicovitch, On Kakeya's Problem and a Similar One, Math. Z. 27, 312-320, 1928.
- Abram Besicovitch, The Kakeya Problem, American Mathematical Monthly, Vol. 70 (1963), pp 697-706
- Charles Fefferman, The multiplier problem for the ball, Ann. of Math. 94 (1971), 330-336.
- Julius Pal, Ueber ein elementares variationsproblem, Kgl. Danske Vid. Selsk. Math.-Fys. Medd., 2:1-35, 1920.
- Terence Tao, From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis and PDE, Notices of the AMS, Vol. 48, no 3 (March 2001), pp 294-303.
- Thomas Wolff, Recent work connected with the Kakeya problem, in Prospects In Mathematics, H. Rossi, ed., AMS 1999.