Norm rigidity and equality cases for the Dyn--Farkhi inequality

For a convex body $K\subset\mathbb{R}^2$ that is symmetric with respect to the origin, and for a nonempty set $S\subset\mathbb{R}^2$, we study the $K$-Hausdorff distance from convex hull, defined by \begin{align*} d^{(K)}(S):=\sup_{x\in \text{conv}(S)}\inf_{s\in S}\|x-s\|_K, \end{align*} where $\|\cdot \|_K$ is the norm whose closed unit ball is $K$. We consider the problem of characterizing the origin symmetric convex bodies $K$ for which \begin{align*} d^{(K)}(A+B)^2\leq d^{(K)}(A)^2+d^{(K)}(B)^2 \end{align*} holds for all nonempty compact $A,B\subset\mathbb{R}^2$. We solve this problem, proving that this property holds if and only if $K$ is an ellipse centered at $0$. We then characterize the conditions for equality for this bound when $K$ is an ellipse.