  In this paper we determine the stretch factor of the $L_1$-Delaunay and
$L_\infty$-Delaunay triangulations, and we show that this stretch is
$\sqrt{4+2\sqrt{2}} \approx 2.61$. Between any two points $x,y$ of such
triangulations, we construct a path whose length is no more than
$\sqrt{4+2\sqrt{2}}$ times the Euclidean distance between $x$ and $y$, and this
bound is best possible. This definitively improves the 25-year old bound of
$\sqrt{10}$ by Chew (SoCG '86). To the best of our knowledge, this is the first
time the stretch factor of the well-studied $L_p$-Delaunay triangulations, for
any real $p\ge 1$, is determined exactly.
