  We consider various inequalities for polynomials, with an emphasis on the
most fundamental inequalities of approximation theory. In the sequel a key role
is played by the generalized Minkowski functional \alpha(K,x), already being
used by Minkowski and contemporaries and having occurred in approximation
theory in the work of Rivlin and Shapiro in the early sixties. We try to
compare real, geometric methods and complex, pluripotential theoretical
approaches, where possible, and formulate a number of questions to be decided
in the future. An extensive bibliography is given to direct the reader even in
topics we do not have space to cover in more detail.
