The monopolist's problem in two dimensions: a seller designs a product menu to maximize profit, facing buyers with two-dimensional types (preferences). The optimal menu induces a partition of buyer space into regions of constant product choice, and the boundary between these regions — the free boundary — encodes the structure of optimal pricing.
The paper establishes that this free boundary, for convex polygonal domains, is a continuous curve of Hausdorff dimension one outside a finite set of singular points. The singularities are isolated: between them, the boundary is locally Hölder continuous (C^α for all α < 1), and under stronger regularity assumptions, locally analytic.
The result connects economic optimization to geometric measure theory. The optimal pricing strategy creates a boundary that is almost everywhere smooth, with singularities confined to a discrete set that can accumulate only at specific geometric features of the domain (corners, cusps). The monopolist's optimal behavior produces geometric regularity — not because the seller aims for smooth boundaries, but because the variational structure of profit maximization enforces regularity through the same mechanisms that control free boundaries in physics (minimal surfaces, phase transitions, obstacle problems).
The economic insight: the monopolist's optimal pricing is geometrically constrained in ways the monopolist doesn't choose and may not recognize. The free boundary's regularity is a theorem about the optimization landscape, not a design choice. Profit maximization on a convex domain forces the pricing boundary to be smooth almost everywhere, and the locations where it can fail to be smooth are determined by the domain's geometry, not the seller's strategy.