In mathematics, the Riesz rearrangement inequality, sometimes called RieszâÂÂSobolev inequality, states that any three non-negative functions , and satisfy the inequality
where , and are the symmetric decreasing rearrangements of the functions , and respectively.
The inequality was first proved by Frigyes Riesz in 1930, and independently reproved by S.L.Sobolev in 1938. Brascamp, Lieb and Luttinger have shown that it can be generalized to arbitrarily (but finitely) many functions acting on arbitrarily many variables.
The Riesz rearrangement inequality can be used to prove the PólyaâÂÂSzegà  inequality.
In the one-dimensional case, the inequality is first proved when the functions , and are characteristic functions of a finite unions of intervals. Then the inequality can be extended to characteristic functions of measurable sets, to measurable functions taking a finite number of values and finally to nonnegative measurable functions.
In order to pass from the one-dimensional case to the higher-dimensional case, the spherical rearrangement is approximated by Steiner symmetrization for which the one-dimensional argument applies directly by Fubini's theorem.
In the case where any one of the three functions is a strictly symmetric-decreasing function, equality holds only when the other two functions are equal, up to translation, to their symmetric-decreasing rearrangements.