The convexification effect of Minkowski summation
Matthieu Fradelizi
Université Paris-Est, Marne-la-Vallée, FranceMokshay Madiman
University of Delaware, Newark, USAArnaud Marsiglietti
California Institute of Technology, Pasadena, USAArtem Zvavitch
Kent State University, Kent, USA
Abstract
Let us define for a compact set the sequence
It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that approaches the convex hull of in the Hausdorff distance induced by the Euclidean norm as goes to . We explore in this survey how exactly approaches the convex hull of , and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on , the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence , and both the Hausdorff distance and effective standard deviation are eventually monotone (once exceeds ). Along the way, we obtain new 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), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.
Cite this article
Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, Artem Zvavitch, The convexification effect of Minkowski summation. EMS Surv. Math. Sci. 5 (2018), no. 1/2, pp. 1–64
DOI 10.4171/EMSS/26