"Let $V$ be an analytic subvariety of a domain $\Omega$ in $\mathbb{C}^n$. When does $V$ have the property that every bounded holomorphic function $f$ on $V$ has an extension to a bounded holomorphic function on $\Omega$ with the same norm? If $\Omega$ is very nice, for example the ball, then this can only happen under very rigid conditions. $V$ must be a holomorphic retract of $\Omega$, i.e. there must exact a holomorphic $r: \Omega \to V$ so that $r|_V = {\rm id}$. Being a retract is always sufficient (as $f \circ r$ gives the extension), but without some convexity condition on $\Omega$ it is not necessary. We shall discuss isometric extensions, and why convexity plays a rôle. This is joint work with Jim Agler and Lukasz Kosinski. "