4 1. Cones
the vector orderings and cones are in one-to-one correspondence. That is,
every cone K of a vector space X induces a vector ordering on X whose
positive vectors are precisely the vectors in JC. If we wish to emphasize that
the vector order on X is generated by the cone /C, then we shall denote it
by /c-
Similarly, if is a vector pre-ordering on a vector space X, then the set
of positive vectors X+ = {x G X: x 0} is a wedge. Conversely, every
wedge W of X gives rise to a vector pre-ordering o n I (by letting x y
if x — y G W) that satisfies X+ — W. Again, if there is a need to emphasize
that the vector pre-ordering on X is generated by the wedge W, then we
shall denote it by yy. In summary, cones give rise to vector orderings and
wedges to vector pre-orderings.
It turns out that wedges are simply sums of vector subspaces with cones.
The proof of the next result is straightforward and is omitted.
Lemma 1.4. A nonempty convex subset W of a vector space is a wedge if
and only if it is the sum of a vector subspace and a cone, that is, if and only
if there exists a vector subspace V and a cone JC such that W = V + JC.
Moreover, whenever W is a wedge, we have the following.
(1) IfweletV = Wn(-W) and JC = {0}U(W\V), thenV is a vector
subspace and JC is a cone satisfying W — V + JC.
(2) If W = V + JC, where V is a vector subspace and K is a cone such
that V n K = {0}, then V = Wn (-W).
(3) The wedge W = V + K is a cone, i.e., V = {0}, if and only if W
contains no straight line passing through the origin.
Now let W be a wedge of a vector space X, and let x be any vector in
the vector subspace generated by W. Choose vectors xi,... , xn G W and
real numbers a i , . . . , an such that x = Y17=i
aixi-
^
w e
^ ^i
= {^: ai
^ 0}
and I2 = {i: a^ 0}, then we have
n
X = ^^
aiXi =
^2
aiXi
~
^2/{~ai)Xi
^W ~W •
i=l ieh ieh
Prom this, it easily follows that the vector subspace generated by W is
precisely W — W.
Definition 1.5. A wedge W of a vector space X is said to be generating if
X — W — W, i.e., if the vector subspace generated by W coincides with X.
It should be clear that a wedge is generating if and only if it majorizes
the space. That is, we have the following simple result.
