In mathematics, the Riesz rearrangement inequality (sometimes called Riesz-Sobolev inequality) states that for any three non-negative functions
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where
History
The inequality was first proved by Frigyes Riesz in 1930, and independently reproved by S.L.Sobolev in 1938. It can be generalized to arbitrarily (but finitely) many functions acting on arbitrarily many variables.
Applications
The Riesz rearrangement inequality can be used to prove the Pólya–Szegő inequality.
One-dimensional case
In the one-dimensional case, the inequality is first proved when the functions
Higher-dimensional case
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.
Equality cases
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.