Optimal quantization in an L-shaped region

Formal definition

• ContainerL = ([0,2] × [0,1]) ∪ ([0,1] × [1,2]), boundary included, with total area 3
• SubmissionExactly n distinct points in L, with coordinates of at most nine decimal places
• AssignmentEvery location is assigned to its nearest point in Euclidean distance; equidistant boundaries have measure zero and do not affect the integral
• ScoreE(P) is the integral of squared distance to the nearest submitted point over the L-shaped region; the verifier integrates in exact rationals and rounds upward at 10⁻¹⁸$$E\left(P\right)=\underset{L}{\int \int }\underset{i}{\min }{\Vert x-p_i\Vert }^{2}dx$$
• ObjectiveMinimize E(P) over all valid point sets P$$\underset{P}{\min }E\left(P\right)$$