According to Wikipedia,
Withinhave subtle aspects to their definition which are contrary to intuition. For example, a line L which is completely contained in the boundary of a polygon P is not considered to be contained in P. This quirk can be expressed as "Polygons do not contain their boundary". This issue is caused by the final clause of the Contains definition above: "at least one point of the interior of B lies in the interior of A". For this case, the predicate Covers has more intuitive semantics (see definition), avoiding boundary considerations.
If Polygons do not Contain their boundaries, do linestrings ever "contain" points?