Lacunary polynomials in $L^1$: geometry of the unit sphere

Let $\mathrm{\Lambda }$ be a finite set of nonnegative integers, and let $\mathcal P(\Lambda)$bethelinearhullofthemonomials$z^k$with$k\in\Lambda$, viewed as a subspace of $L^1$ on the unit circle. We characterize the extreme and exposed points of the unit ball in $\mathcal{P}(\mathrm{\Lambda })$.