Page 174 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Thecaseofdimension n 161

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Definition 6.6 (i) For A, B ⊂ R , n ≥ 1,wedefine
A + B = {a + b : a ∈ A, b ∈ B} .

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(ii) For x ∈ R and A ⊂ R ,welet
d (x, A)= inf {|x − a| : a ∈ A} .

Example 6.7 (i) If n =1, A =[a, b], B =[c, d],wehave
A + B =[a + c, b + d] .

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(ii) If we let B R = {x ∈ R : |x| <R}, we get
B R + B S = B R+S .

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Proposition 6.8 Let A ⊂ R , n ≥ 1 be compact and B R = {x ∈ R : |x| <R}.
The following properties then hold.
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(i) A + B R = {x ∈ R : d (x, A) ≤ R}.
(ii) If A is convex, then A + B R is also convex.

Proof. (i) Let x ∈ A + B R and
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X = {x ∈ R : d (x, A) ≤ R} .
We then have that x = a + b for some a ∈ A and b ∈ B R , and hence

|x − a| = |b| ≤ R

which implies that x ∈ X.Conversely, since A is compact, we can find, for every
x ∈ X, an element a ∈ A so that |x − a| ≤ R.Letting b = x − a, we have indeed
found that x ∈ A + B R .
(ii) Trivial.
We now examine the meaning of the proposition in a simple example.

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Example 6.9 If A is a rectangle in R ,we find that A + B R is given by the
figure below. Anticipating, a little, on the following results we see that we have
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M A + B R = M (A)+ RL (∂A)+ R π
where L (∂A) is the perimeter of ∂A.