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 \begin{equation*} A[k] = \left \{a_1+\cdots +a_k: a_1, \ldots , a_k\in A\right \}=\sum _{i=1}^k A \end{equation*} 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 [Concentration, functional inequalities and isoperimetry, Amer. Math. Soc., Providence, RI, 2011] conjectured that the volume of $\frac {1}{k}A[k]$ is nondecreasing 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 [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 185–189] that this conjecture holds true if $d=1$ but fails for any $d \geq 12$. 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 $d \geq 7$.