Do Minkowski averages get progressively more convex?

Let us define, for a compact set $A\subset {\mathbb{R}}^n$, the Minkowski averages of $A$: $$A(k)=\{\frac{a_1+\cdots +a_k}{k}:a_1,\dots ,a_k\in A\}=\frac{1}{k}\left(\right)\mathrm{.}$$ We study the monotonicity of the convergence of $A(k)$ towards the convex hull of $A$, when considering the Hausdorff distance, the volume deficit and a non-convexity index of Schneider as measures of convergence. For the volume deficit, we show that monotonicity fails in general, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider’s non-convexity index, we prove that a strong form of monotonicity holds, and for the Hausdorff distance, we establish that the sequence is eventually nonincreasing.