We study the Hausdorff distance to convex hull, which for a compact set A ⊂ R n A Rⁿ is defined by d (A) ≔ d H (A, c o n v (A) ) d (A) ≔dH (A, conv (A) ), where d H dH is the Hausdorff metric. In 2004, Dyn and Farkhi Numer. Funct. Anal. Optim. 25 (2004), pp. 363–377 conjectured that d 2 d² is subadditive on compact sets in R n Rⁿ. 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 R².
Mark Meyer (Wed,) studied this question.