  Let us define, for a compact set $A \subset \mathbb{R}^n$, the Minkowski
averages of $A$: $$ A(k) = \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldots,
a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A +
\cdots + A}}\Big). $$ 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.
