A sharp $p$-subadditive bound for the $l_p$ Hausdorff distance from convex hull

We study the $l_p$ Hausdorff distance from convex hull of a compact set $A\subset\mathbb{R}^n$, which is the distance \begin{equation*} d^{(l_p)}(A):=\sup_{x\in conv(A)}\inf_{a\in A}\|x-a\|_p, \end{equation*} where $\|\cdot\|_p$ is the $l_p$-norm on $\mathbb{R}^n$. We prove that when $n=2$ and $1\leq p<\infty$, the function $(d^{(l_p)})^p$ is subadditive with respect to Minkowski summation, up to multiplication by the factor $\max\{1,2^{p-2}\}$, and we observe that this bound is sharp.