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In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions f : R n → R + {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} , g : R n → R + {\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} and h : R n → R + {\displaystyle h:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} satisfy the inequality
where f ∗ : R n → R + {\displaystyle f^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} , g ∗ : R n → R + {\displaystyle g^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} and h ∗ : R n → R + {\displaystyle h^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} are the symmetric decreasing rearrangements of the functions f {\displaystyle f} , g {\displaystyle g} and h {\displaystyle h} respectively.
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