Let $A \subset B$, and let $Y$ be a Banach function space on $B$. Two common

interpolation problems are:

(i) Given $f : A \to \C$, find $g \in Y$ so that $g|_A = f$ and $g$ has the smallest possible norm.

(ii) Let $X$ be a Banach function space on $A$. Determine if there is a constant $M$, and if so what the best value is, so that for every $f: A \to \C$ in $X$ there exists $g \in Y$ so that $g |_A = f$ and $\| g \|_Y \leq M \| f \|_X$.

It is often the case in Problem (i) that the optimal solution has extra regularity properties. We consider an analogous question for Problem (ii).

Suppose $B$ is a pseudoconvex set in $\C^n$, and $A$ is an analytic subvariety. Let $X$ and $Y$ be the bounded holomorphic functions on $A$ and $B$ respectively. If the constant $M$ can be chosen to be $1$, what does this say about the relationship between $A$ and $B$?

If $B$ is nice, eg. the ball, then $A$ is rigidly constrained: it must be a holomorphic retract. But {\em every } $A$ can occur for some $B$.

We shall discuss both the nice and non-restricted cases. The talk is base on joint work with J. Agler and \L. Kosi\'nski.