  We study first-passage percolation on $\mathbb Z^d$, $d\ge 2$, with
independent weights whose common distribution is compactly supported in
$(0,\infty)$ with a uniformly-positive density. Given $\epsilon>0$ and
$v\in\mathbb Z^d$, which edges have probability at least $\epsilon$ to lie on
the geodesic between the origin and $v$? It is expected that all such edges lie
at distance at most some $r(\epsilon)$ from either the origin or $v$, but this
remains open in dimensions $d\ge 3$. We establish the closely-related fact that
the number of such edges is at most some $C(\epsilon)$, uniformly in $v$. In
addition, we prove a quantitative bound, allowing $\epsilon$ to tend to zero as
$\|v\|$ tends to infinity, showing that there are at most
$O\big(\epsilon^{-\frac{2d}{d-1}}(\log \|v\|)^C\big)$ such edges, uniformly in
$\epsilon$ and $v$. The latter result addresses a problem raised by
Benjamin-Kalai-Schramm (2003).
  Our technique further yields a strengthened version of a lower bound on
transversal fluctuations due to Licea-Newman-Piza (1996).
