  Let $d\in\mathbb N$, $\alpha\in\mathbb R$, and let $f :\mathbb R^d\setminus
\{0\} \rightarrow (0,\infty)$ be locally Lipschitz and positively homogeneous
of degree $\alpha$ (e.g. $f$ could be the $\alpha$th power of a norm on
$\mathbb R^d$). We study a generalization of the Eden model on $\mathbb Z^d$
wherein the next edge added to the cluster is chosen from the set of all edges
incident to the current cluster with probability proportional to the value of
$f$ at the midpoint of this edge, rather than uniformly. This model is
equivalent to a variant of first passage percolation where the edge passage
times are independent exponential random variables with parameters given by the
value of $f$ at the midpoint of the edge.
  We prove that the $f$-weighted Eden model clusters have an a.s. deterministic
limit shape if $\alpha< 1$, which is an explicit functional of $f$ and the
limit shape of the standard Eden model, and estimate the rate of convergence to
this limit shape. We also prove that if $\alpha>1$, then there is a norm $\nu$
on $\mathbb R^d$ (depending on $\alpha$) such that if we set $f(z) = \nu(z)^{
\alpha}$, then the $f$-weighted Eden model clusters are a.s.\ contained in a
Euclidean cone with opening angle $<\pi$ for all time. We further show that
there does \emph{not} exist a norm on $\mathbb R^d$ for which this latter
statement holds for all $\alpha>1$; and that there is no choice of function $f$
for which the above statement holds with $\alpha=1$.
  Our basic approach is to compare the local behavior of the $f$-weighted first
passage percolation to that of unweighted first passage percolation with iid
exponential edge weights (which is equivalent to the unweighted Eden model).
  We include a list of open problems and several computer simulations.
