In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions
f :
n
satisfy the inequality where
are the symmetric decreasing rearrangements of the functions
The inequality was first proved by Frigyes Riesz in 1930,[1] 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.
[2] The Riesz rearrangement inequality can be used to prove the Pólya–Szegő inequality.
In the one-dimensional case, the inequality is first proved when the functions
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.
[3] 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.
[4] 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.