14

The Euclidean isoperimetric problem

In this chapter we study the **Euclidean isoperimetric problem**,

characterizing its minimizers as the Euclidean balls of measure *m.* Taking into account that *P*(*B*) = * _{n}*, this result will be equivalent to proving the following

**Theorem 14.1** *If E is a Lebesgue measurable set in with* |*E*| < , *then*

*Equality holds if and only if* |*E**B*(*x*, *r*)| = 0 *for some* *x* , *r* > 0.

**Remark 14.2** When ...

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