Consider the "infinite chessboard" on the plane. Think of it as the lattice $X_1:=\mathbb{Z}^2$, and also finer chessboards $X_n$ corresponding to $\frac{1}{n}\cdot \mathbb{Z}^2$, $n\geq 1$. Given two squares (i.e. vertices) $u,v$ of $X_n$ one can define the "knight distance" $d_n(u,v)$ as the minimum number of moves that a "knight" (moving as in the usual game of chess) must do to get from $u$ to $v$, divided by $n$.

Now take two points $a,b$ on the plane $\mathbb{R}^2$, and define their "knight distance at step $n$" to be the minimum of $d_n(u,v)$ for $u,v\in X_n$ such that $d_E(u,a)$ and $d_E(v,b)$ are minimal (where $d_E$ is the Euclidean distance on $\mathbb{R}^2$).

Define the "knight distance" on the plane by $d_K(a,b):=\lim_{n\to\infty}d_n(a,b)$.

Does it define an actual distance (metric) on $\mathbb{R}^2$? Assuming it does, how does the spheres look like for this metric? Any interesting properties? Is there a self-homeomorphism of $\mathbb{R}^2$ that pullbacks $d_K$ to $d_E$?

(This was a question I happened to ask myself at high school -and never thought really to answer- of which I was reminded of by just reading this MO thread by Joseph O'Rourke. Provided it makes sense, it still looks like a legitimate question to me...)