  The metric $D_\alpha (q,q')$ on the set $Q$ of particle locations of a
homogeneous Poisson process on $R^d$, defined as the infimum of $(\sum_i |q_i -
q_{i+1}|^\alpha)^{1/\alpha}$ over sequences in $Q$ starting with $q$ and ending
with $q'$ (where $| . |$ denotes Euclidean distance) has nontrivial geodesics
when $\alpha > 1$. The cases $1 <\alpha < \infty$ are the Euclidean
first-passage percolation (FPP) models introduced earlier by the authors while
the geodesics in the case $\alpha = \infty$ are exactly the paths from the
Euclidean minimal spanning trees/forests of Aldous and Steele. We compare and
contrast results and conjectures for these two situations. New results for $1 <
\alpha < \infty$ (and any $d$) include inequalities on the fluctuation
exponents for the metric ($\chi \le 1/2$) and for the geodesics ($\xi \le 3/4$)
in strong enough versions to yield conclusions not yet obtained for lattice
FPP: almost surely, every semi-infinite geodesic has an asymptotic direction
and every direction has a semi-infinite geodesic (from every $q$). For $d=2$
and $2 le \alpha < \infty$, further results follow concerning spanning trees of
semi-infinite geodesics and related random surfaces.
