Volume of the Minkowski sums of star-shaped sets

Abstract

For a compact set $A\subset {\mathbb{R}}^d$ and an integer $k\ge 1$, let us denote by $$A[k]=\left\{a_1+\cdots +a_k:a_1,\dots ,a_k\in A\right\}=\sum _{i=1}^{k}A$$ the Minkowski sum of $k$ copies of $A$. A theorem of Shapley, Folkmann and Starr (1969) states that $\frac{1}{k}A[k]$ converges to the convex hull of $A$ in Hausdorff distance as $k$ tends to infinity. Bobkov, Madiman and Wang (2011) conjectured that the volume of $\frac{1}{k}A[k]$ is non-decreasing in $k$, or in other words, in terms of the volume deficit between the convex hull of $A$ and $\frac{1}{k}A[k]$, this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch (2016) that this conjecture holds true if $d=1$ but fails for any $d\ge 12$. In this paper we show that the conjecture is true for any star-shaped set $A\subset {\mathbb{R}}^d$ for $d=2$ and $d=3$ and also for arbitrary dimensions $d\ge 4$ under the condition $k\ge (d-1)(d-2)$. In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in ${\mathbb{R}}^d$, for any $d\ge 7$.