- LOG SOBOLEV INEQUALITY VIA ISOPERIMETRIC INEQUALITY 207
In this section we prove a sharp form of the logarithmic Sobolev inequal-
ity, relating to the isoperimetric inequality. The method of proof is to apply
spherical symmetrization (also called Steiner symmetrization) and
to reduce the inequality to (22.89). The following result is due to Perelman;
see also Proposition 4.1 in [139] by one of the authors for the proof in theIn= Cn case.
THEOREM 22.16 (Logarithmic Sobolev via isoperimetric inequality). Let
(Mn, g) be a complete Riemannian manifold which satisfies an isoperimetric
inequality; that is, there exists a constant In E (0, oo) such that(22.90) (Area(on)t 2: In (Vol(n)r-^1
for any compact domain n c M whose boundary is C^1. Then on (M, g) we
have the logarithmic Sobolev inequalityJM (2IV7~l2 - ~2 log~2) dμ +log (JM ~2dμ) JM ~2 dμ
(22.91) 2: (Sn+ log (~:))JM ~2
dμ,where~ is any W^1 •^2 function on M, Sn~~ log(27r) + n, and Cn = nnwn is
the isoperimetric constant of Euclidean space ~n.
REMARK 22.17. A general idea relating isoperimetric inequalities and
Sobolev inequalities is to consider the level sets of functions as the bound-
aries of the superlevel sets of these functions. Technically this is facilitated
by the co-area formula, which holds for Lipschitz functions.
PROOF. Observe that the isoperimetric inequality (22.90) is invariantunder scalings of the metric g. For the rescaled metric g ~
see that (22.91) is equivalent to^7
(22.92)( )
2/n
~: g, we2 (~) 2/n JM IV~l2 dP,-JM ~2 log (~2) dP, +log (JM ~2 dP,) JM ~2 dP,
2: Sn JM ~2
dj),,where dP, denotes the volume form of g.
By replacing~ by l~I and using an approximation argument, it suffices
to prove (22.92) under the assumption that ~ is a nonnegative function
in C^1 with compact support. More precisely, we may first approximate a
(^7) This follows easily from the facts that if g ~ ( 7,;')^2 /n g, then
dP, = ~ dμ and JV'lf 12 ~ JV'lf J~ = ( ~ )-
2