Polarization, sign sequences and isotropic vector systems

Abstract

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).

Additive colourful Carathéodory type results with an application to radii

“…Again, we must have points from U 1 (say u 1 2 ) and from U 2 (say u 2 2 ) inside the other arc determined by ±u 3 1 . Again, for the same reason, U i ∩ relint(B(±u 3 1 , π/3)) = ∅, i = 1, 2. Assuming that u 3 1 = e 1 , Proposition 1.1(4) implies that U i ∩ {x ∈ S 1 : x 2 ≤ 0} = ∅, i = 1, 2.…”

“…Proposition 1.1(4)). Assuming that this second point in this arc is u 3 2 , we furthermore observe that it must necessarily belong to the arc of S 1 determined by −u 3 1 and the left-most vertex of B(u 2 1 , π/3). Since u 3 1 lies in S 1 between e 1 and the right-most vertex of B(u 2 1 , π/3), we see that the angle between u 3 2 (as well as u 1 1 ) and the left-most vertex of B(u 2 1 , π/3) is at most π/3.…”

“…Proof of Corollary 1.1. Since R(K i ) = 1 with K i ⊂ B 2 2 , let us denote by u j i ∈ K j ∩ S 1 , for i, j ∈ [3], the points given in Proposition 1.1 (3). Denoting by…”

“…It does not seem easy to guess the optimal bounds in extending Theorem 1.3 to arbitrary n ∈ N and i ≥ n + 1. Nevertheless, we were told by Alexandr Polyanskii [18] (see also [3]) that the following conjecture for i = n + 1 seems reasonable. Then max l1,...,ln+1…”

“…All vectors in the gray region, together with u 2 Since 0 is contained in the convex hull of U 3 there must be a point u3 2 on the green arc.…”