It is proven that for any system of n points z1, . . . , zn on the (complex) unit circle, there exists another point z of norm 1, such thatTwo proofs are presented: one uses a characterisation of equioscillating rational functions, while the other is based on Bernstein's inequality.

Let Ξ 0 = [−1, 1], and define the segments Ξ n recursively in the following manner: for every n = 0, 1, . . . , let Ξ n+1 = Ξ n ∩ [a n+1 − 1, a n+1 + 1], where the point a n+1 is chosen randomly on the segment Ξ n with uniform distribution. For the radius ρ n of Ξ n we prove that n(ρ n − 1/2) converges in distribution to an exponential law, and we show that the centre of the limiting unit interval has arcsine distribution.

We determine the order of magnitude of the nth ℓp-polarization constant of the unit sphere S d−1 for every n, d 1 and p > 0. For p = 2, we prove that extremizers are isotropic vector sets, whereas for p = 1, we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for d = 2, we discuss the optimality of equally spaced configurations on the unit circle.2010 Mathematics Subject Classification. 52A40(primary), and 31C20(secondary).