  We prove consistency of four different approaches to formalizing the idea of
minimum average edge-length in a path linking some infinite subset of points of
a Poisson process. The approaches are (i) shortest path from origin through
some $m$ distinct points; (ii) shortest average edge-length in paths across the
diagonal of a large cube; (iii) shortest path through some specified proportion
$\delta$ of points in a large cube; (iv) translation-invariant measures on
paths in $\Reals^d$ which contain a proportion $\delta$ of the Poisson points.
We develop basic properties of a normalized average length function $c(\delta)$
and pose challenging open problem
