Fix $A \in \mathbb{R}^{m\times m}, c>0, r>0.$ Consider the function $f: \mathbb{R}_+ \times \mathbb{R}^m \to \mathbb{R}$ such that, for all $x \in B_r(0)$, it holds that $$f(t, e^{At}\cdot x) = \left(1-\frac{1}{c \ \ r}\ \mathrm{dist}(x,B_r(0)) \right)^+.$$
1) What does this function look like when plotted, for example, with $m=2$?
2) What does it represent?
Notation: $B_r(0)$ is the ball with center $0$ and radius $r$, and $\mathrm{dist}(x,B_r(0))$ denotes the distance from $x$ to $B_r(0)$.