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.

Analogous sets also exist in higher dimensions.

The existence of such a set as N was first proved in 1927, by Polish mathematician Otto M. Nikodym. The existence of higher-dimensional Nikodym sets was first proved in 1986, by British mathematician Kenneth Falconer.

Nikodym sets are closely related to Kakeya sets (also known as Besicovitch sets).

The existence of Nikodym sets is sometimes compared with the Banach–Tarski paradox. There is, however, an important difference between the two: the Banach–Tarski paradox relies on non-measurable sets.

References

• Falconer, K. J. (1986), "Sets with prescribed projections and Nikodym sets", Proceedings of the London Mathematical Society, S3-53: 48–64, doi:10.1112/plms/s3-53.1.48.