Space-time tradeoff and the best algorithm Consider some language $L$ such that:

$L \in DTIME(O(f(n))) \cap DSPACE(O(g(n)))$

and so that

$L \not\in DTIME(o(f(n))) \cup DSPACE(o(g(n)))$

In other words, the fastest machine $M$ computes $L$ in time $O(f(n))$ and the most space efficient machine $M&#39;$ computes $L$ while using space $O(g(n))$. 

What can be said about the space efficiency of M or the time efficiency of M'? Or more precisely, if $\mathbb{M}_T$ is the set of all machines that compute $L$ in $O(f(n))$ then what can we say about the most space efficient machine in $\mathbb{M}_T$? What about the same thing for the obvious space version: $\mathbb{M}_S$.

Alternatively, can $f(n)$ and $g(n)$ be used to define some good space-time tradeoffs? Under what conditions is $TS \in o(f(n)g(n))$ or more generally for some space-time tradeoff $h(T,S)$ under what conditions is $h(T,S) \in h(o(f(n)),o(g(n)))$.
The prototypical f and g here would probably be poly-time and polylog space.  The interesting problem here is connectivity (in directed graphs) which can be solved in polynomial time (using linear space) or in polylog space (using super-polynomial time).  It is a famous open problem whether it can be solved in TIME-SPACE(poly,polylog), a class known as SC.

I.e. your question is a well-known open problem.  I don't think that anything non-trivial is known here.