This paper analyzes the location of a distribution center in an urban area using a single-source Weber problem with continuous piecewise fixed cost to find a global optimal location. The fixed cost is characterized by a Kriging interpolation method. To make the fixed cost tractable, we approximate this interpolation with a continuous piecewise function that is convex in each piece, using Delaunay triangulation. We present a decomposition formulation, a decomposition conic formulation and a conic logarithmic disaggregated convex combination model to optimally solve the single-source Weber problem with continuous piecewise fixed cost. Although our continuous approach does not guarantee the global optimal feasible location, it allows us to delimit a zone where we can intensify the search of feasible points. For instances we tested, computational results show that our continuous approach found better locations than the discrete approach in 23.25% of the instances and that the decomposition formulation is the best one, in terms of CPU time.