How far apart can we be?

Geometry Level 5

We have 15 points {Ai} \{A_i\} placed within the unit sphere. What is the maximum possible value of

1i<j15AiAj2? \sum_{1 \leq i < j \leq 15} \left| A_i A_j\right|^2 ?

Details and assumptions

Each of the points Ai=(xi,yi,zi)R3 A_i = (x_i, y_i, z_i) \in \mathbb{R}^3 satisfy xi2+yi2+zi21 x_i ^2 + y_i ^2 + z_i ^2 \leq 1 .

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