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.