1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

250 6. ENTROPY AND NO LOCAL COLLAPSING

w h ere N * -- N-2N 2 an d A N -- ( r(N/r(N) 2 ) ) 2/N 7rN(N-^1 2 ). B S y ter l" mg ' s £ orrnu^1 a,

I' (N) '"" ..j2iiNN-~e-N, so that AN rv 7r;N for N large, where I' is the

Garnrna function. Hence, if JJRN IFl^2 dx = 1, then by above two inequalities

log (AN LN l\7 Fl

2

dx) 2: ~ log (LN IFIN dx)

2: ~ LN (log IFI) IFl

2

dx,

where we used ~ (N - 2) = fr.

Given f : ]Rn --+JR with JJRn j2dx = 1, let N = n£ and define F : JRN --+JR

by

JI,

F (x) =';: IT! (xk),

k=l

where x = (x1, ... 'xe)' Xk E ]Rn fork= 1, ... 'e. Now

\7F(x) = (\7f(x1) \7f(xe))

F(x) f(x1) , ... , f(xe) ·

Hence

and

JI,

since fJRn f (x)^2 dx = 1. Using fJRnf F (x)^2 dx = IT fJRn f (xk)^2 dxk = 1, we
k=l
have