Analytic and Probabilistic Problems in Discrete Geometry

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence u1, . . . , un of norm 1 vectors in a real Hilbert space H , there exists a unit vector v ∈ H , such that X 1 〈ui, v〉2 6 n . The 2-dimensional case is proved by complex analytic methods. For the higher dimensional extremal cases, we prove a tensorisation result that is similar to F. John’s theorem about characterisation of ellipsoids of maximal volume. From this, we deduce that the only full dimensional locally extremal system is the orthonormal system. We also obtain the same result for the weaker, original polarization problem. The second chapter investigates a problem in probabilistic geometry. Take n independent, uniform random points in a triangle T . Convex chains between two fixed vertices of T are defined naturally. Let Ln denote the maximal size of a convex chain. We prove that the expectation of Ln is asymptotically α n1/3, where α is a constant between 1.5 and 3.5 – we conjecture that the correct value is 3. We also prove strong concentration results for Ln, which, in turn, imply a limit shape result for the longest convex chains.