Additive properties of the Hausdorff metric

For a compact set $A \subset \mathbb{R}^n$ , the Hausdorff distance to convex hull is defined by $d(A) := d_H(A, conv(A))$, where $d_H$ is the Hausdorff metric. A notable conjecture in this area is the Dyn–Farkhi conjecture, which posits that $d_2$ is subadditive on compact sets in $\mathbb{R}^n$ . In 2018, Fradelizi, Madiman, Marsiglietti and Zvavitch disproved the Dyn–Farkhi conjecture when $n ≥ 3$, but the question remained open in $\mathbb{R}^2$ . Recent work has found that the Dyn– Farkhi conjecture is an affirmative in $\mathbb{R}^2$ . In this talk, I will outline the proof that $d_2$ is subadditive on compact sets in $\mathbb{R}^2$.