Page 178 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Thecaseofdimension n 165

This is a direct consequence of the fact that the logarithm function is concave
and hence à ! à !
n n n
X X Y
log ≥ λ i log u i =log u λ i .
λ i u i
i
i=1 i=1 i=1
Step 2. Let F be the family of all open sets A of the form
n
Y
A = (a i ,b i ) .
i=1
We will now prove the theorem for A, B ∈ F. We will even show that for every
λ ∈ [0, 1], A, B ∈ F we have
1/n 1/n 1/n
[M (λA +(1 − λ) B)] ≥ λ [M (A)] +(1 − λ)[M (B)] . (6.2)
The theorem follows from (6.2) by setting λ =1/2.If we let
n n
Y Y
A = (a i ,b i ) and B = (c i ,d i )
i=1 i=1
we obtain
n
Y
λA +(1 − λ) B = (λa i +(1 − λ) c i ,λb i +(1 − λ) d i ) .
i=1
Setting, for 1 ≤ i ≤ n,
b i − a i d i − c i
u i = ,v i = (6.3)
λ (b i − a i )+ (1 − λ)(d i − c i ) λ (b i − a i )+(1 − λ)(d i − c i )
we find that
λu i +(1 − λ) v i =1, 1 ≤ i ≤ n, (6.4)
n n
M (A) Y M (B) Y
= u i , = v i . (6.5)
M (λA +(1 − λ) B) M (λA +(1 − λ) B)
i=1 i=1
We now combine (6.1), (6.4) and (6.5) to deduce that
n
n
λ [M (A)] 1/n +(1 − λ)[M (B)] 1/n Y 1/n Y 1/n
= λ u +(1 − λ) v
1/n i i
[M (λA +(1 − λ) B)]
i=1 i=1
n n
X u i X v i
≤ λ +(1 − λ)
n n
i=1 i=1
n
1 X
= (λu i +(1 − λ) v i )= 1
n
i=1