Page 180 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Thecaseofdimension n 167

We let
± ± ± ±
A = A ∩ A µ and B = B ∩ B ν
µ ν
provided these intersections are non empty; and we deduce that
M ± N ±
[ [
±
±
A = A ± and B = B .
±
µ ν
µ=1 ν=1
+
By construction we have M + <M and M − <M,while N , N − ≤ N.If
+
−
λ ∈ [0, 1],we see that λA +(1 − λ) B + and λA +(1 − λ) B − are separated
by {x : x i = λa +(1 − λ) b} and thus
¡ + + ¢ ¡ ¢
M (λA +(1 − λ) B)= M λA +(1 − λ) B + M λA +(1 − λ) B − .
−
+
Applying the hypothesis of induction to A , B + and A , B ,wededucethat
−
−
h i n
£ ¡ + ¢¤ 1/n £ ¡ + ¢¤ 1/n
M (λA +(1 − λ) B) ≥ λ M A +(1 − λ) M B
h i n
£ ¡ ¢¤ 1/n £ ¡ ¢¤ 1/n
+ λ M A − +(1 − λ) M B − .
Using (6.7) we obtain
+ h
M (A ) 1/n 1/n i n
M (λA +(1 − λ) B) ≥ λ [M (A)] +(1 − λ)[M (B)]
M (A)
− h
M (A ) 1/n 1/n i n
+ λ [M (A)] +(1 − λ)[M (B)] .
M (A)
The identity (6.6) and the above inequality imply then (6.2).
Step 4. We now show (6.2) for any compact set, concluding thus the proof
of the theorem. Let > 0, we can then approximate the compact sets A and B,
by A and B as in Step 3, so that

|M (A) − M (A )| , |M (B) − M (B )| ≤ , (6.8)

|M (λA +(1 − λ) B) − M (λA +(1 − λ) B )| ≤ . (6.9)
Applying (6.2) to A , B , using (6.8) and (6.9), we obtain, after passing to the
limit as → 0,the claim
1/n 1/n 1/n
[M (λA +(1 − λ) B)] ≥ λ [M (A)] +(1 − λ)[M (B)] .