  We study first-passage percolation through related optimization problems over
paths of restricted length. The path length variable is in duality with a shift
of the weights. This puts into a convex duality framework old observations
about the convergence of the normalized Euclidean length of geodesics due to
Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results
about geodesic length and the regularity of the shape function as a function of
the weight shift. For points far enough away from the origin, the ratio of the
geodesic length and the $\ell^1$ distance to the endpoint is uniformly bounded
away from one. The shape function is a strictly concave function of the weight
shift. Atoms of the weight distribution generate singularities, that is, points
of nondifferentiability, in this function. We generalize to all distributions,
directions and dimensions an old singularity result of Steele and Zhang for the
planar Bernoulli case. When the weight distribution has two or more atoms, a
dense set of shifts produce singularities. The results come from a combination
of the convex duality, the shape theorems of the different first-passage
optimization problems, and modification arguments.
