Alweiss has observed a lower bound of $\binom{n + 1}{2}$ follows from considering the subset of
$\mathbb{R}^{n + 1}$ formed of all vectors $e_i + e_j$ where $e_i$, $e_j$ are distinct coordinate
vectors. This set can be viewed as a subset of some $\mathbb{R}^n$, and is easily checked to have
the required property.
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