Page 176 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Thecaseofdimension n 163

Example 6.12 If A = B R ,we find the well known formula for the area of the
sphere S R = ∂B R
¡ ¢ ¡ ¢
M B R + B − M B R
L (S R ) = lim
→0
n n
[(R + ) − R ] ω n n−1
= lim = nR ω n
→0
where ω n is as above.
We are now in a position to state the theorem that plays a central role in
the proof of the isoperimetric inequality.
n
Theorem 6.13 (Brunn-Minkowski theorem). Let A, B ⊂ R , n ≥ 1,be
compact, then the following inequality holds
1/n 1/n 1/n
[M (A + B)] ≥ [M (A)] +[M (B)] .
1/n
Remark 6.14 (i) The same proof establishes that the function A → (M (A))
is concave. We thus have

1/n 1/n 1/n
[M (λA +(1 − λ) B)] ≥ λ [M (A)] +(1 − λ)[M (B)]
n
for every compact A, B ⊂ R and for every λ ∈ [0, 1].
(ii) One can even show that the function is strictly concave. This implies
that the inequality in the theorem is strict unless A and B are homothetic.
Example 6.15 Let n =1.
(i) If A =[a, b], B =[c, d],wehave A + B =[a + c, b + d] and
M (A + B)= M (A)+ M (B) .

(ii) If A =[0, 1], B =[0, 1] ∪ [2, 3],we find A + B =[0, 4] and hence

M (A + B)= 4 >M (A)+ M (B)= 3.
We will prove Theorem 6.13 at the end of the section. We are now in a
position to state and to prove the isoperimetric inequality.
n
Theorem 6.16 (Isoperimetric inequality). Let A ⊂ R , n ≥ 2,be a com-
pact set, L = L (∂A), M = M (A) and ω n be as above, then the following
inequality holds
n
n
L − n ω n M n−1 ≥ 0 .
Furthermore equality holds, among all convex sets, if and only if A is a ball.