On the Brunn-Minkowski theorem (Q1107122)

Let \(K_ 1\) and \(K_ 2\) be n-dimensional convex bodies which satisfy for some \(\alpha\in]0,1[\) and some \(\epsilon \in [0,+\infty [\) the inequality \[ V(\alpha K_ 1+(1-\alpha)K_ 2)^{1/n}\leq \alpha V(K_ 1)^{1/n}+(1-\alpha)V(K_ 2)^{1/n}+\epsilon \quad, \] where V denotes the volume. The problem investigated in the present paper is whether \(K_ 1\) and \(K_ 2\) are in some sense homothetic. Several theorems that answer this question associated with the Brunn-Minkowski inequality are proved.