Innocent-looking program transformations can easily change the space
complexity of lazy functional programs. The theory of \emph{space
improvemement} seeks to characterise those local program
transformations which are guaranteed never to worsen asymptotic space
complexity of any program. Previous work by the authors introduced the
space improvement relation and showed that a number of simple local
transformation laws are indeed space improvements. This paper seeks an
answer to the following questions: is the improvement relation inhabited
by interesting program transformations, and, if so, how might they be
established? We show that the asymptotic space improvement relation is
semantically badly behaved, but that the theory of \emph{strong space
improvement} posesses a fixed-point induction theorem which permits
the derivation of improvement properties for recursive definitions.
With the help of this tool we explore the landscape of space improvement by
considering a range of classical program transformations.