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

TL;DR

This paper characterizes the extreme and exposed points of the unit ball in a subspace of $L^1$ generated by lacunary polynomials, enhancing understanding of their geometric structure.

Contribution

It provides a complete characterization of the extreme and exposed points of the unit ball in spaces of lacunary polynomials within $L^1$, a novel geometric insight.

Findings

Identifies all extreme points of the unit ball in $ ext{span}\{z^k : k ext{ in } ext{finite set}",

Provides a description of exposed points in these polynomial subspaces

Enhances understanding of the geometry of lacunary polynomial spaces in $L^1$

Abstract

Let $\Lambda$ be a finite set of nonnegative integers, and let $\mathcal P(\Lambda)$ be the linear hull of the monomials $z^k$ 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 $\mathcal P(\Lambda)$.