Let $A$ be an open bounded subset of euclidean $n$-space $\mathbb{R}^n$. For $x\in A$, let $r=r(x)$ be the maximal radius such that the ball centred at $x$ with radius $r=r(x)$ is contained in $A$, i.e. $r(x)$ is the maximal $r>0$ satisfying $B_r(x) \subset A$. N.B. the radius $r(x)$ coincides with distance-to-the-boundary $dist(x,\partial A)$.

For $x\in A$, let $M_x$ be the maximal ball containing $B_{r(x)}(x)$ and contained in $A$. Thus $M_x$ is the maximal ball satisfying $B_{r(x)}(x) \subset M_x \subset A$.

My question: If we assume $A$ is open and bounded, then does $x\mapsto M_x$ vary continuously with $x$?

N.B. For given $x$, I argue that there exists a unique maximal ball $M_x$ satisfying the above conditions.

Remark. For my purposes a positive answer would imply that the centre $m$ of $M_x$ varies continuously with $x$. Thus we would obtain a continuous map $x\mapsto m(x)$ from $x$ to the centre $m(x)$ of the max ball $M_x$.

Remark. I remember reading an article/book of Vitali Kapovich which had some similar constructions, but I cannot recall the specifics. My goal is to establish continuity of $x\mapsto m(x)$ with mild hypotheses on $A$, i.e. without requiring a $C^1$ boundary.

Remark. There is a variational characterization of the max-centre defined as the argmax of a convex function on a convex domain. Therefore the argmax is unique element on the boundary of the domain, and varies continuously with respect to the datum.