Extremal metrics on toric surfaces, I

The datum σ yields a measure dσ on the boundary ∂P—on each edge E we take dσ to be a constant multiple of the standard Lebesgue measure with the constant normalised so that the mass of the edge is σ(E). Equally, the datum σ specifies an affine-linear defining function λE for each edge E, i.e. the edge lies in the hyperplane λ E (0). We choose an inward-pointing normal vector v at a point of E with |iV dμ| = dσE where dμ is the fixed standard area form on R and we specify λE by the condition that ∇vλE = 1. For a continuous function f on P we set