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

For a compact set A in Rⁿ the Hausdorff distance from A to conv (A) is defined by equation* d (A): =₀ (₀) ₗ ₀|x-a|, equation* where for x= (x₁, , xₙ) ⁿ we use the notation |x|=x₁²++xₙ². It was conjectured in 2004 by Dyn and Farkhi that d² is subadditive on compact sets in Rⁿ. In 2018 this conjecture was proved false by Fradelizi et al. when n3. The conjecture can also be verified when n=1. In this paper we prove the conjecture when n=2 and in doing so we prove an interesting representation of the sumset conv (A) +conv (B) for full dimensional compact sets A, B in R².