Abstract
A finite subset X of the unit sphere \({\mathbb{S}^{d-1}}\) in \({\mathbb{R}^d}\) is called extremal if, for every \({x \in X}\), there is a hemisphere that contains \({X \setminus \{x\}}\) in its interior and has x on its boundary. Let P denote the probability that a random sample of d + 1 points, chosen uniformly from \({\mathbb{S}^{d-1}}\), is extremal. We show that \({P = 1-(d + 2)/2^{d}}\).
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Dedicated to Tibor Bisztriczky, Gábor Fejes Tóth, and Endre Makai, Jr., on the occasion of their 70th birthdays
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Maehara, H., Martini, H. An analogue of Sylvester’s four-point problem on the sphere. Acta Math. Hungar. 155, 479–488 (2018). https://doi.org/10.1007/s10474-018-0814-y
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DOI: https://doi.org/10.1007/s10474-018-0814-y