Linear problems in the space of polynomials of degree at most 3

Denote by $P_{n}, n\in N$ the linear space of real polynomials $p$ of degree at most $n$. There are various ways in which we can introduce norm in $P_{n}$, here the problem is investigated when $||p||=max\{|p(x)|:x\in [-1;1]\}$. Let $B_{n}=\{p\in P_n:||p||\le 1\}$ be the unit ball and let $EB_{n}$ be the set of the extreme points of $B_{n}$, i.e. such points $p\in B_{n}$ that $B_{n}\setminus \{p\}$ is convex. The sets $EB_{0}, EB_{1}$ and $EB_{2}$ are known and it turns out that also $EB_{3}$ has a particularly simple form. In this paper we determine $EB_{3}$ and give some conclusions and applications of the main results. Moreover, several examples are included. The coefficient region for the polynomials of degree exceeding 3 seems very complicated.