The convexification effect of Minkowski summation

  • Matthieu Fradelizi

    Université Paris-Est, Marne-la-Vallée, France
  • Mokshay Madiman

    University of Delaware, Newark, USA
  • Arnaud Marsiglietti

    California Institute of Technology, Pasadena, USA
  • Artem Zvavitch

    Kent State University, Kent, USA
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Let us define for a compact set ARnA \subset \mathbb{R}^n the sequence

A(k)={a1++akk:a1,,akA}=1k(A++Ak times).A(k) = \left\{\frac{a_1+\cdots +a_k}{k}: a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big).

It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that A(k)A(k) approaches the convex hull of AA in the Hausdorff distance induced by the Euclidean norm as kk goes to \infty. We explore in this survey how exactly A(k)A(k) approaches the convex hull of AA, 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 Rn\mathbb R^n, 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 A(k)A(k) 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 AA with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence A(k)A(k), and both the Hausdorff distance and effective standard deviation are eventually monotone (once kk exceeds nn). 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