  Choosing an encoding over binary strings for input/output to/by a Turing
Machine is usually straightforward and/or inessential for discrete data (like
graphs), but delicate -- heavily affecting computability and even more
computational complexity -- already regarding real numbers, not to mention more
advanced (e.g. Sobolev) spaces. For a general theory of computational
complexity over continuous data we introduce and justify QUANTITATIVE
admissibility as requirement for sensible encodings of arbitrary compact metric
spaces, a refinement of qualitative 'admissibility' due to
[Kreitz&Weihrauch'85]:
  An admissible representation of a T0 space $X$ is a (i) continuous partial
surjective mapping from the Cantor space of infinite binary sequences which is
(ii) maximal w.r.t. continuous reduction. By the Kreitz-Weihrauch (aka "Main")
Theorem of computability over continuous data, for fixed spaces $X,Y$ equipped
with admissible representations, a function $f:X\to Y$ is continuous iff it
admits continuous a code-translating mapping on Cantor space, a so-called
REALIZER. We define a QUANTITATIVELY admissible representation of a compact
metric space $X$ to have (i) asymptotically optimal modulus of continuity,
namely close to the entropy of $X$, and (ii) be maximal w.r.t. reduction having
optimal modulus of continuity in a similar sense.
  Careful constructions show the category of such representations to be
Cartesian closed, and non-empty: every compact $X$ admits a linearly-admissible
representation. Moreover such representations give rise to a tight quantitative
correspondence between the modulus of continuity of a function $f:X\to Y$ on
the one hand and on the other hand that of its realizer: the MAIN THEOREM of
computational complexity. This suggests (how) to take into account the
entropies of the spaces under consideration when measuring algorithmic cost
over continuous data.
