## The Spherical Isoperimetric Inequality

Here is an application of the spherical isoperimetric inequality.

FactYou can’t cut up a beach ball into equal parts with a path that is too short.

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## BPM 1.4.2 (Sp 84)

Let for . We show that is monotonically increasing. First we rewrite . Taking derivatives, and doing some algebra, we get . We wish to show that everywhere. This would follow from . However, this follows from the convexity of . Which says that for all .

We compute . Taking we see that . We then have that . Since is monotonically increasing, and positive, we have that as . We compute . Applying concavity, we have for all . We then check that , where is by applying . It follows that .

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