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Solution of Scott's Problem on the Number of Directions Determined by a Point Set in 3-Space

Let P be a set of n points in R3, not all in a common plane. We solve a problem of Scott (1970) by showing that the connecting lines of P assume at least 2n - 5 different directions if n is odd and at least 2n - 7 if n is even. The bound for odd n is sharp.