A weakened version of completeness for abelian topological groups.

An abelian topological group has the quasi-convex compactness property (qcp) if the quasi-convex envelope of a compact set is again compact. In the class of locally quasi-convex groups the qcp turns out to be a weakened form of completeness.
We shall study the qcp in conjunction with other properties. For instance, in a metrizable locally quasi-convex group the qcp and completeness are equivalent. Furthermore, a metrizable MAP group with the qcp must be locally quasi-convex. Concerning the hereditary behavior of the quasi-convex compactness property, if is a subgroup of a topological abelian group with the qcp which is closed in the -modification of , then also has the qcp. In the other direction, if is a strongly dually embedded subgroup of and has the qcp, then is closed in the -modification of .
If the starting group has additionally a topological vector space structure, an analogue of a well-known theorem of Krein also holds for . Nevertheless, we shall present an example of a complete metrizable locally quasi-convex group which does not have the qcp when endowed with the weak topology..