Nikodym set

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. pp. p. 100. ISBN 0-521-25694-1.  MR867284