We study the Schneider non-convexity index of compact sets $A\subset \mathbb{R}^{n}$ , defined to be the smallest $\lambda >0$ such that the sumset $A+\lambda \operatorname{conv}(A)$ is convex. We compute a sharp lower bound on the index of the Minkowski sum of two compact sets in $\mathbb{R}$ , and establish a family of fractional subadditive inequalities for sums of m compact sets in $\mathbb{R}^{n}$ .
