Oscar Ortega Moreno: An optimal plank problem

Abstract: We give a new proof of Fejes Toth's zone conjecture: for any sequence v_1,v_2,...,v_n of unit vectors in a real Hilbert space H, there exists a unit vector v in J such that |(v_k, v)| >= sin( \pi / 2n) for all k. This can be seen as a sharp version of the plank theorem for real Hilbert spaces. Our approach is inspired by K. Ball's solution to the complex plank problem and thus unifies both the complex and the real solution under the same method.