Optimal quantization in the unit square

Formal definition

• ContainerThe unit square [0, 1]², with lower-left corner at (0, 0) and upper-right corner at (1, 1)
• SubmissionExactly n distinct points, each coordinate written as a decimal string with at most nine decimal places
• AssignmentEvery location belongs to its nearest submitted point (the Voronoi partition); equidistant boundaries have measure zero and do not affect the integral
• ScoreE(P) is the integral over the unit square of the squared distance to the nearest submitted point (equal to the mean since the square has area 1); the verifier computes the exact rational integral and rounds upward at 10⁻¹⁸$$E\left(P\right)=\int \int \underset{i}{\min }{\Vert x-p_i\Vert }^{2}dx$$
• ObjectiveMinimize E(P) over all valid point sets P$$\underset{P}{\min }E\left(P\right)$$