  We analyze the geometrical structure of the passage times in the last passage
percolation model. Viewing the passage time as a piecewise linear function of
the weights we determine the domains of the various pieces, which are the
subsets of the weight space that make a given path the longest one. We focus on
the case when all weights are assumed to be positive, and as a result each
domain is a pointed polyhedral cone. We determine the extreme rays, facets, and
two-dimensional faces of each cone, and also review a well-known simplicial
decomposition of the maximal cones via the so-called order cone. All geometric
properties are derived using arguments phrased in terms of the last passage
model itself. Our motivation is to understand path probabilities of the
extremal corner paths on boxes in $\Z^2$, but all of our arguments apply to
general, finite partially ordered sets.
