In mathematics, a Nikodym set is the seemingly paradoxical result of a construction in measure theory. A Nikodym set in the unit square S in the Euclidean plane E² is a subset N of S such that

* the area (i.e. two-dimensional Lebesgue measure) of N is 1;

* for every point x of N, there is a straight line through x that meets N only at x.

The existence of such a set was N was first proved in 1927 by the Polish mathematician Otton M. Nikodym. Nikodym sets are closely related to Kakeya sets (also known as Besicovitch sets).

See also

* Banach-Tarski paradox

References

* Falconer, Kenneth J. (1986). The geometry of fractal sets, Cambridge Tracts in Mathematics 85. Cambridge: Cambridge University Press, p. 100. ISBN 0-521-25694-1. MR867284