## 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.

Let’s take a closed compact surface with a Riemannian metric, which induces an inner metric . Let be the *inner diameter* and , where is a closed loop. The loops in the infimum are *bisecting curves* which split the sphere into two regions of equal area.

Theorem 1The round unit sphere has .

The round unit sphere is nothing other than . The equator is clearly a bisecting curve and . The sphere has diameter . This shows that . Now we need to show that we hit the equality. The spherical isoperimetric inequality says:

Theorem 2For a simple smooth curve on let be the smaller of the two regions it encloses and let be its length. We have:

Suppose that divides the sphere into two regions of equal area. We have

and thus . This inequality says that if we want to enclose half the area of the sphere then we can’t do better than using of length. That means that the equator was optimal and thus .

Of Waists and Spheres | Parker Glynn-Adeysaid, on 2013/11/19 at 21:54[…] a way to a simple but important case of the waist inequality which I’ve mentioned before. The classical isoperimetric inequality on the sphere says: Any curve of length on the sphere […]