On the Brunn-Minkowski theorem

Abstract

Let K ₁ and K ₂ be n-dimensional convex bodies. If V denotes volume the Brunn-Minkowski theorem in its simplest form states that $V(K_1 + K_2 )^{1/n} \geqslant V(K_1 )^{1/n} + V(K_2 )^{1/n}$, and that equality holds if and only if K ₁ and K ₂ are homothetic. We consider the following associated stability problem: If $V(K_1 + K_2 )^{1/n}$ differs not more than ε from $V(K_1 )^{1/n} + V(K_2 )^{1/n}$ how close is K ₁ (in terms of the Hausdorff metric) to a nearest homothetic copy of K ₂? Several theorems that answer questions of this kind are proved. These results can also be expressed as inequalities that are stronger than the original Brunn-Minkowski inequality. Furthermore, some consequences concerning stability statements for other geometric inequalities are discussed.