This is an announcement for the paper “On the monotonicity of Minkowski sums towards convexity” by Matthieu Fradelizihttps://arxiv.org/find/math/1/au:+Fradelizi_M/0/1/0/all/0/1, Mokshay Madimanhttps://arxiv.org/find/math/1/au:+Madiman_M/0/1/0/all/0/1, Arnaud Marsigliettihttps://arxiv.org/find/math/1/au:+Marsiglietti_A/0/1/0/all/0/1, Artem Zvavitchhttps://arxiv.org/find/math/1/au:+Zvavitch_A/0/1/0/all/0/1.
Abstract: Let us define for a compact set $A\subset\mathbb{R}_n$ the sequence $$ A(k)={\frac{a_1+\cdots+a_k}{k}: a_1, \cdots, a_k\in A}=\frac{1}{k}(A+\cdots+ A). $$ By a theorem of Shapley, Folkman and Starr (1969), $A(k)$ approaches the convex hull of $A$ in Hausdorff distance as $k$ goes to $\infty$. Bobkov, Madiman and Wang (2011) conjectured that Vol$_n(A(k))$ is non-decreasing in $k$, where Vol$_n$ denotes the $n$-dimensional Lebesgue measure, or in other words, that when one has convergence in the Shapley-Folkman-Starr theorem in terms of a volume deficit, then this convergence is actually monotone. We prove that this conjecture holds true in dimension $1$ but fails in dimension $n\geq 12$. We also discuss some related inequalities for the volume of the Minkowski sum of compact sets, showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex. Then we consider whether one can have monotonicity in the Shapley-Folkman-Starr theorem when measured using alternate measures of non-convexity, including the Hausdorff distance, effective standard deviation or inner radius, and a non-convexity index of Schneider. For these other measures, we present several positive results, including a strong monotonicity of Schneider's index in general dimension, and eventual monotonicity of the Hausdorff distance and effective standard deviation. Along the way, we clarify the interrelationships between these various notions of non-convexity, demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.05486