  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 $u_1,\dots, u_n$ of norm 1 vectors in a real Hilbert
space $\mathscr H$, there exists a unit vector $v \in \mathscr H$, such that $$
\sum \frac{1}{\langle u_i, v \rangle^2} \leq n^2. $$ 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 $L_n$ denote the maximal size
of a convex chain. We prove that the expectation of $L_n$ is asymptotically
$\alpha \, n^{1/3}$, where $\alpha$ is a constant between 1.5 and 3.5 -- we
conjecture that the correct value is 3. We also prove strong concentration
results for $L_n$, which, in turn, imply a limit shape result for the longest
convex chains.
