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):=a(A)infx∈ A|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 n≥3. 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.