Solving the Minkowski problem by continuation from the unit sphere: spherical harmonic discretization and error estimates
Given a positive function $ρ$ on the unit sphere $\mathbb{S}^2$, the classical Minkowski problem asks for a closed convex surface in $\mathbb{R}^3$ with Gauss curvature $1/ρ(ν)$ at the point with outer normal $ν$. In terms of the support function $h$, it is the Monge--Ampère equation $\det(\nabla^2h+hσ)=ρ$ on $\mathbb{S}^2$, uniquely solvable up to translation when $ρ$ has zero first moment. We solve it by continuation from the sphere: along the linear path $ρ_t=(1-t)+tρ$, the solution $h(t)$ evolves in the artificial time $t$, and its velocity $\partial_th$ solves a linear elliptic equation in divergence form whose kernel consists of the translations. In time we use the explicit Euler or trapezoidal method with step $τ$; in space, spherical harmonics of degree at most $L$ with a Lagrange multiplier for the kernel. For $ρ\in H^{s+2}(\mathbb{S}^2)$ with real $s\ge4$, we prove that all numerical solutions stay strictly convex, with $H^1$ errors $O(τ+L^{-(s+1)})$ for Euler and $O(τ^3+L^{-(s+1)})$ for the trapezoidal scheme, including data quadrature, under independent thresholds for $τ$ and $L$. The proof rests on a priori estimates for the perturbed equation satisfied by each iterate, which keep every step uniformly elliptic, and on the exactly quadratic determinant, which raises the trapezoidal order from two to three. The continuation thus supplies a good initial guess for Newton's method: from the final state it converges quadratically to the Galerkin solution, itself within $O(L^{-(s+1)})$ of the exact solution, whereas started directly from the sphere its first undamped step left the convex cone in our ellipsoid test. Experiments illustrate the rates, use the multiplier as a compatibility diagnostic, reconstruct convex polyhedra from facet normals and areas, and locate umbilics with indices.