2019/10/14 by Matthieu Fradelizi, Fradelizi, Matthieu, Zsolt Lángi +3 · 1 citation
Computer Science · Mathematics · #52A38 #52A40 #60E15 #Computational Geometry and Mesh Generation #FOS: Mathematics #Functional Analysis (math.FA) #Limits and Structures in Graph Theory #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · doi:10.48550/arxiv.1910.06146
openalex publication_date 2019/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a compact set A ⊂ \mathbb Rd and an integer k≥ 1, let us denote by A[k] = \a1+⋯ +ak: a1, …, ak∈ A\=∑i=1k A the Minkowski sum of k copies of A. A theorem of Shapley, Folkmann and Starr (1969) states that (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 (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 (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 ≥ 12. In this paper we show that the conjecture is true for any star-shaped set A ⊂ \mathbb Rd for d=2 and d=3 and also for arbitrary dimensions d ≥ 4 under the condition k ≥ (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 Rd, for any d ≥ 7.