By Beer [Topologies on Closed and Closed Convex Sets,
Kluwer, 1993, p. 28], gaps and excesses can be expressed
in terms of the induced surroundings
S_{r} = { (x, y) \in X^{2} : d(x, y) < r } , with r > 0 ,
in a metric space X(d).
To do the same with diameters, for any A \subseteq X,
I have tried to use the quantity
d(A) = 2 inf { r>0 : \exists x\in X : A \subseteq S_{r}(x) } .
However, I could only prove the inequality diam(A) \le d(A) .
Of course, all the above mentioned results have to be geralized
to some generalized metrics and metric-type relators.
The equality does not hold in general. For example, if $X$ be a discrete metric space with more than 1 point, then one gets $d(X)=2$ while $diam X=1$. (As I understood, $S_{r}(x)$ is the usual open ball with radius $r$ and centrum in $x$?) Maybe you should add some conditions on $X$, for example $X$ is a convex subset of a normed linear space or geodesic metric space.
Dear Taras Radul,
Thank you for your example and remarks.
The discrete metric was also mentioned by
my colleague Zoltan Boros.
However, the quantity
r(A)=\inf { r>0: \exists x\in X: A\subseteq S_{r}(x) }
may still be a useful tool in a metric space X.
A paper of mine [A common generalization of postman,
radial, and river metrics], becoming by now too difficult for me,
may also contain some relevant examples.
With best wishes,
Arpad Szaz
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