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I want to know if there is a proper term for the following kind of points, which I call it center(borrowed from graph theory).

Let $(X,d)$ be a metric space. $x\in X$ is a center of $A\subset X$ if $x\in A$ and $$ \sup_{a\in A} d(x,a) = \inf_{b\in A} \sup_{a\in A} d(b,a) $$.

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up vote 2 down vote accepted

It is called a Chebyshev center of the set (using indefinite article since such a point is not unique in general). Unfortunately, the beginning of the Wikipedia article is a mess: having nonempty interior is irrelevant, and the part after "alternatively" defines a different concept. But Google will confirm that the term Chebyshev center normally describes exactly the thing you introduced.

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