In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Ruziewicz problem (sometimes Banach-Ruziewicz problem) in measure theory asks whether the usual Lebesgue measure

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

 on the n-sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 is characterised, up to proportionality, by its properties of being finitely additive, invariant under rotation

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s, and defined on all Lebesgue measurable sets.

This was answered affirmatively and independently for n ≥ 4 by Grigory Margulis

Grigory Margulis

Gregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...

 and Dennis Sullivan

Dennis Sullivan

Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...

 around 1980, and for n = 2 and 3 by Vladimir Drinfel'd

Vladimir Drinfel'd

Vladimir Gershonovich Drinfel'd is a Ukrainian and Soviet mathematician at the University of Chicago.The work of Drinfeld related algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the...

 (published 1984). It fails for the circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

.

The name is for Stanisław Ruziewicz.