Smallest ratio of largest to smallest distance

Formal definition

• ContainerUnconstrained in the plane: the distance ratio is invariant under translation and scaling, and configurations are submitted inside the unit square [0, 1]²
• SubmissionExactly n distinct points
• Extreme distancesd(P) is the minimum pairwise distance and D(P) is the maximum pairwise distance$$d\left(P\right)=\underset{i<j}{\min }\Vert p_i-p_j\Vert ,D\left(P\right)=\underset{i<j}{\max }\Vert p_i-p_j\Vert$$
• Scoreρ(P) is invariant under translation, rotation, and nonzero scaling; internally, the verifier compares squared distances in exact arithmetic and rounds ρ(P) upward to nine decimal places$$\rho \left(P\right)=\frac{D\left(P\right)}{d\left(P\right)}$$
• ObjectiveMinimize ρ(P) over all n-point configurations P$$\underset{P}{\min }\rho \left(P\right)$$