Approximate compactness

A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a limit point $y\in M$. Approximate compactness of a given set ensures the existence of an element of best approximation for any $x\in X$. The concept of approximate compactness was introduced [1] in connection with the study of Chebyshev sets (cf. Chebyshev set) in a Banach space, which made it possible to describe convex Chebyshev sets in certain spaces. In fact, let $X$ be a uniformly-convex smooth Banach space. For a Chebyshev set $M\subset X$ to be convex, it is necessary and sufficient that it be approximately compact. It follows, in particular, that the set of rational fractions with fixed degrees of the numerator and the denominator is not a Chebyshev set in the space $L_p$ ($1<p<\infty$) if the degree of the denominator is not smaller than one [1].

For subsequent studies on this subject see [2].

References

[1]	N.V. Efimov, S.B. Stechkin, "Approximative compactness and Čebyšev sets" Soviet Math. Dokl. , 2 : 5 (1961) pp. 522–524 Dokl. Akad. Nauk SSSR , 140 : 3 (1961) pp. 522–524
[2]	A.L. Garkavi, "The theory of best approximation in normed linear spaces" Itogi Nauk. Mat. Anal. 1967 (1969) pp. 75–132 (In Russian)