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Let $X$ be a subset of $ \mathbb{R}^n $ , such that given $x,y \in X $ we have $ \lVert x-y\rVert = 1 $. Prove that $ |X| < n+2 $.

This result is geometrically obvious, but how can I prove it for any $n$?

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2 Answers 2

up vote 2 down vote accepted

Try translating one point to the origin and then showing the others are linearly independent.

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WLOG one member of $X$ is $0$. If $x_1, \ldots, x_{n+1}$ are $n+1$ other members of $X$, consider the Gramian matrix $M$ with entries $x_i \cdot x_j$. Note that $M_{ii} = 1$ and $M_{ij} = 1/2$ for $i \ne j$. Show that the eigenvalues of $M$ are $1/2$ and $(n+1)/2$, and since $M$ is nonsingular, $x_1, \ldots, x_n$ must be linearly independent.

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