AlphaEvolve problems·alphaevolve:50
Max to min ratios
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Let $n,d \\geq 2$. Let $C(d,n)$ denote the largest quantity such that, given any $n$ distinct points $x_1,\\dots,x_n$ in $\\R^d$, the maximum distance $\\max_{1 \\leq i < j \\leq n} \\|x_i-x_j\\|$ between the points is at least $C(d,n)$ times the minimum distance $\\min_{1 \\leq i < j \\leq n} \\|x_i-x_j\\|$. Establish upper and lower bounds for $C(d,n)$. What are the configurations that attain the minimal ratio between the two distances?
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qed.bot, “Max to min ratios”, https://qed.bot/t/alphaevolve-50, as of 30 Sep 2026.