Lacunary polynomials in \(L^1\): geometry of the unit sphere (Q2656123)

Let \(\mathcal{P}_N\) be the set of complex polynomials of degree at most \(N.\) Suppose that \(k_1< k_2< \ldots< k_M<N\) are positive integers. Consider the set \(\Lambda:=\{0,1,\ldots, N\} \setminus \{k_1, k_2, \ldots, k_M\}.\) Denote by \(\mathcal{P} (\Lambda)\) the space of complex polynomials of the form \(\sum_{k\in \Lambda} c_k z^k\) viewed as a subspace of \(L^1(\mathbb{T)},\) where \(\mathbb{T}:=\{z \in \mathbb{C} : \ |z|=1\}.\) In the paper the author obtains the full description of the extreme and exposed points of the unit ball in \(\mathcal{P}(\Lambda)\) in terms of the rank of a certain matrix \(\mathcal{M}\) constructed therein.