Page 179 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

166 Isoperimetric inequality

and hence the result.
Step 3. We now prove (6.2) for any A and B of the form
M N
[ [
A = A µ ,B = B ν
µ=1 ν=1
where A µ , B ν ∈ F, A ν ∩A µ = B ν ∩B µ = ∅ if µ 6= ν. The proof is then achieved
through induction on M + N. Step2hasprovedthe result when M = N =1.
We assume now that M> 1.We then choose i ∈ {1,..., n} and a ∈ R such that
if
+ n − n
A = A ∩ {x ∈ R : x i >a} ,A = A ∩ {x ∈ R : x i <a}
+
then A and A contain at least one of the A µ , 1 ≤ µ ≤ M, i.e. the hyperplane
−
{x i = a} separates at least two of the A µ (see Figure 7.2).

Figure 6.2: separating hyperplane

We clearly have
¡ + ¢ ¡ ¢
M A + M A − = M (A) . (6.6)
We next choose b ∈ R (such a b exists by an argument of continuity) so that if

+ n − n
B = B ∩ {x ∈ R : x i >b} ,B = B ∩ {x ∈ R : x i <b}
then
+
+
M (A ) M (B ) M (A ) M (B )
−
−
= and = . (6.7)
M (A) M (B) M (A) M (B)