The Dyn–Farkhi conjecture and the convex hull of a sumset in two dimensions

We study the Hausdorff distance to convex hull, which for a compact set A ⊂ R n A⊂ R^n is defined by d ( A ) ≔ d H ( A , c o n v ( A ) ) d(A)≔d_H(A,conv(A)) , where d H d_H is the Hausdorff metric. In 2004, Dyn and Farkhi [Numer. Funct. Anal. Optim. 25 (2004), pp. 363–377] conjectured that d 2 d^2 is subadditive on compact sets in R n R^n . In 2018, Fradelizi, Madiman, Marsiglietti, and Zvavitch [EMS Surv. Math. Sci. 5 (2018), pp. 1–64] found a counterexample to this conjecture when n ≥ 3 n≥ 3 . In this paper, we resolve the Dyn–Farkhi conjecture when n = 2 n=2 . In doing so, we prove a new representation of the sumset c o n v ( A ) + c o n v ( B ) conv(A)+conv(B) for compact sets A , B ⊂ R 2 A,B⊂ R^2 .