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Let $C_{d}(\\tau_{1}, ..., \\tau_{n}) \\subset \\RR^{d}$ denote the cyclic polytope of dimension $d$ with $n$ vertices $(\\tau_{1},\\tau_{1}^{2},...,\\tau_{1}^{d}), ..., (\\tau_{n},\\tau_{n}^{2},...,\\tau_{n}^{d})$. We are interested in finding the smallest integer $\\gamma_{d}$ such that if $\\tau_{i+1} - \\tau_{i} \\geq \\gamma_{d}$ for $1 \\leq i < n$, then $C_{d}(\\tau_{1}, ..., \\tau_{n})$ is normal. One of the known results is $\\gamma_{d} \\leq d (d + 1)$. 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