*Definition [#qf31c3a0]
-Let K be a convex subset of a linear topological space X. We say that K is line-compact if L∩K is a compact subset of L (in its unique linear topology) for every line L in X.

*Property [#jce8e8fc]
-if K is σ-convex and [[line-closed]], K is line-compact.

*Reference [#s68142ff]
-Rosenthal, Haskell, ''L^1-convexity. Functional analysis'', 156--174, Lecture Notes in Math., 1332, Springer, Berlin, 1988.