Exact asymptotics of the optimal $L_{p,\Omega}$-error of linear spline interpolation

In this paper we provide the exact asymptotics of the optimal weighted $L_p$-error, $0<p< \infty$, of linear spline interpolation of $C^2$ functions with positive Hessian. The full description of the behavior of the optimal error leads to the algorithm for construction of an asymptotically optimal sequence of triangulations. In addition, we compute the minimum of the $L_p$-error of linear interpolation of the function $x^2+y^2$ over all triangles of unit area for all $0<p<\infty$. This provides the exact constant in the asymptotics of the optimal error.