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.