Metrics on compact spaces compatible with entropy properties

Preprint series: 99-44, Reports on Analysis

MSC:
54E45 Compact (locally compact) metric spaces
52C99 None of the above but in this section

Abstract: Given a compact metric space $(X,d)$ so-called controllable coverings
of $X$ are defined by a uniformity condition in terms of entropy numbers.
This geometric condition is closely related to Kolmogoroff's entropy
function of $X$.

We discuss the following question motivated by a problem from approximation
theory: Let $d^\prime$ be another metric on $X$ such that $(X,d^\prime)$ is
compact. When do $d$ and $d^\prime$ admit the same controllable coverings?
We present sufficient as well as necessary conditions by the aid of
Kolmogoroff's entropy function and, in the end, give a construction for a
new metric $d^\prime$ essentially different from the original $d$ which
gives rise to the same controllable coverings.

Keywords: copact metric space, covering, entropy number, Kolmogoroff's entropy function