Isoperimetry and functional inequalities

1.1 Brunn-Minkowski inequality 1.1 Theorem. (Brunn-Minkowski, ’88) If A and B are non-empty compact sets then for all λ ∈ [0, 1] we have vol ((1− λ)A+ λB) ≥ (1− λ)(volA) + λ(volB). (B-M) Note that if either A = ∅ orB = ∅, this inequality does not hold since (1−λ)A+λB = ∅. We can use the homogenity of volume to rewrite Brunn-Minkowski inequality in the form vol (A+B) ≥ (volA) + (volB). (1.1) We can deduce from this inequality the isoperimetric inequality. 1.2 Theorem. Among sets with prescribed volume, the Euclidean balls are the one with minimum surface area. Proof. We can assume that C is compact and volC = volB 2 . We have