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 $\varepsilon>0$ there is $\delta>0$ such that the inequalities $\|x\pm y\|<1+\delta$ imply, for $y\in\mathcal X$, that $\|y\|<\varepsilon$. We are concerned with certain subspaces of $H^\infty$, 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 $H^\infty$. It is well known that the strongly extreme points of $\text{ball}(H^\infty)$ are precisely the inner functions, while the (usual) extreme points of this ball are the unit-norm functions $f\in H^\infty$ with $\log(1-|f|)$ non-integrable over the circle. Here we show that similar characterizations remain valid for our perturbed $H^\infty$-type spaces. Also, we investigate to what extent a non-inner function can differ from a strongly extreme point.