Is the difference of distance function convex?

Question

Suppose all the points are in $\mathbb{R}^2$. Given two points $z_1,z_2$ and a number A, is the solution set of the following inequality function convex? $$ f(x)=d(x,z_1)-d(x,z_2)-A \leq 0 $$ where $d(\cdot,\cdot)$ is a 2-dimensional distance function, such as $L_2$-norm or $L_1$-norm. Or does the solution set of inequality will be convex when the distance function satisfies some properties?

Answer 1

Hint: If $A=0$ then it is convex , otherwise It might be non-convex , Try examples in $\Bbb R^2$