Directions of recession of a set
constitute a cone that we denote
.
We introduce the
notation
The
,
if not empty, constitutes a subspace. We call it a "linearity space" of
.
To see that the closedness is necessary consider the set
,
see the figure (
Closedness and
recession
). The only candidate for the direction of recession is
.
However, the point
translates outside of
along
.
To see that the requirement
is necessary consider the sets
and
for
,
see the figure (
Closedness and
recession
). These do not intersect but have a common direction of
recession.
To see that the closedness is necessary consider
and
.
The intersection is
.
It has a direction of recession
.
The
has no direction of recession.
Note that the compactness of
is important. In absence of compactness we cannot state that the
is closed and we cannot state that
.
