Volume of the Minkowski sums of star-shaped sets

TL;DR

This paper investigates the monotonicity of volume in Minkowski sums of star-shaped sets, proving the conjecture in low dimensions and under certain conditions, while providing counterexamples in higher dimensions.

Contribution

It extends the understanding of volume monotonicity for Minkowski sums of star-shaped sets, proving the conjecture for dimensions 2 and 3, and for higher dimensions with conditions, and presents counterexamples for generalizations.

Findings

Volume of scaled Minkowski sums is non-decreasing for star-shaped sets in dimensions 2 and 3.

The conjecture holds for arbitrary dimensions when the number of summands exceeds a certain bound.

Counterexamples show the conjecture does not hold universally in higher dimensions for sums of distinct sets.

Abstract

For a compact set $A \subset {\mathbb R}^d$ and an integer $k\ge 1$, let us denote by $$ A[k] = \left\{a_1+\cdots +a_k: a_1, \ldots, a_k\in A\right\}=\sum_{i=1}^k A$$ 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 $d \geq 12$. 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 $d \geq 7$.