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

Abstract

For a compact set $A$ in ${\mathbb{R}}^n$ the Hausdorff distance from $A$ to $\text{conv}(A)$ is defined by \begin{equation*} d(A):=\sup_{a\in\text{conv}(A)}\inf_{x\in A}|x-a|, \end{equation*} where for $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ we use the notation $|x|=\sqrt{x_1^2+\cdots +x_n^2}$. It was conjectured in 2004 by Dyn and Farkhi that $d^2$ is subadditive on compact sets in ${\mathbb{R}}^n$. In 2018 this conjecture was proved false by Fradelizi et al. when $n\ge 3$. 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 $\text{conv}(A)+\text{conv}(B)$ for full dimensional compact sets $A,B$ in ${\mathbb{R}}^2$.