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

TL;DR

This paper proves the Dyn-Farkhi conjecture for the case of two dimensions, demonstrating that the squared Hausdorff distance to the convex hull is subadditive in this setting, and provides a new representation of sumsets in the plane.

Contribution

The paper establishes the conjecture in 2D and introduces a novel representation of the sumset of convex hulls for full-dimensional compact sets in the plane.

Findings

Proved the Dyn-Farkhi conjecture for n=2.

Derived a new representation of the sumset of convex hulls in 2D.

Confirmed the conjecture's validity specifically in two-dimensional space.

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 $|x|=\sqrt{x_1^2+\dots+x_n^2}$. 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 $n\geq3$. 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 $\mathbb{R}^2$.