How far apart can we be?

Geometry Level 5

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

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

Details and assumptions

Each of the points \( A_i = (x_i, y_i, z_i) \in \mathbb{R}^3\) satisfy \( x_i ^2 + y_i ^2 + z_i ^2 \leq 1 \).

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