Inner functions as strongly extreme points: stability properties

Given a Banach space $\mathcal{X}$, let $x$ be a point in $\text{ball}(\mathcal{X})$, the closed unit ball of $\mathcal{X}$. One says that $x$ is a strongly extreme point of $\text{ball}(\mathcal{X})$ if it has the following property: for every $\epsilon >0$ there is $\delta >0$ such that the inequalities $\mathrm{\parallel }x\pm y\mathrm{\parallel }<1+\delta$ imply, for $y\in \mathcal{X}$, that $\mathrm{\parallel }y\mathrm{\parallel }<\epsilon$. We are concerned with certain subspaces of $H^{\mathrm{\infty }}$, the space of bounded holomorphic functions on the disk, that arise upon imposing finitely many linear constraints and can be viewed as finite-dimensional perturbations of $H^{\mathrm{\infty }}$. It is well known that the strongly extreme points of $\text{ball}(H^{\mathrm{\infty }})$ are precisely the inner functions, while the (usual) extreme points of this ball are the unit-norm functions $f\in H^{\mathrm{\infty }}$ with $\log (1-\mathrm{\mid }f\mathrm{\mid })$ non-integrable over the circle. Here we show that similar characterizations remain valid for our perturbed $H^{\mathrm{\infty }}$-type spaces. Also, we investigate to what extent a non-inner function can differ from a strongly extreme point.