As people have said, obviously in general you don't have A contained in the ball of diameter diamA, more clearly r=diam A/2. What you can have immediately is A contained in the ball of of radius diam A, i.e. r= diamA.
In the proof of Evans - Gariepy, they use Steiner symmetry lemma, meaning that for any set A Lebesgue measurable, you take any line d. You try to make the set to be symmetric w.r.t. the line d but keep the same volume (this can be make by using Fubini's theorem). It's not so hard to show that, by doing that procedure, the volume remains the same while diamA decreases (philosophically it's correct since make somethings more round, more nice will decrease the diameter).
After using Steiner symmetry lemma for the n axis e_1,...,e_n, you will have the new object, name A' that is symmetric w.r.t. e_1,...,e_n and |A|=|A'| while diamA' le diamA. Notice that A' need not to be the sphere, it can have some diamond shape for example. But, the nice thing is A' subset B(0,diamA'/2).
So, we get the result.
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