C Geometric tools
Louis Dupaigne · 2011
We begin our discussion with three fundamental inequalities that are deeply related to the geometry of RN . Theorem C.1.1 Let N ≥ 2. Among all sets Ω ⊂ RN having a smooth boundary of given finite (N−1) Hausdorff measure, the ball has the largest N-dimensional volume, that is, |Ω| N−1N ≤ C |∂Ω| , (C.1) with C = B1 N−1N /∂ B1. Equality holds if and only if Ω is the ball of radius r = |∂Ω||∂ B1| 1 N−1 .