6

1. Cones

Full sets that are not convex

Clearly, the intersection of a family of full sets is a full set. From this,

we easily see that every subset B of L is included in a smallest (with respect

to inclusion) full subset—called the full hull of B and denoted [B]. A

moment's thought reveals that

[B] = (B + L+)n(B-L+) = | J [x,y\.

x,yeB

From the first formula we conclude that

(1) the full hull of a convex set is convex, and

(2) the full hull of a circled set is

circled.5

A subset A of an ordered vector space L is said to be bounded above

if there exists some vector x (called an upper bound of A) such that a x

holds true for all a G A. Similarly, A is bounded below if there exists

some vector x (called a lower bound of A) satisfying x a for all a G A.

A subset A of L is order bounded if A is bounded from above and below

or, equivalently, if it is included in an order interval.

A vector u in an ordered vector space L is called the least upper bound

(or the supremum) of a nonempty subset A of L, written u = sup A, if

(a) u is an upper bound of A1 i.e., a u holds for all a E i , and

(b) u is the least upper bound of A in the sense that for any upper

bound v of A we have u v.

It should be clear from the above definition that a nonempty subset of an

ordered vector space can have at most one supremum. The definition of the

greatest lower bound (or the infimum) of a nonempty set A, denoted

inf A, is introduced analogously.

The classical lattice notations for the supremum and infimum of a finite

set {xi,... ,x

n

} are

sup{xi,... ,x

n

} = \J Xi and inf{xi,. ..,x

n

} = f\ xi.

2 = 1 1 = 1

We also write sup{x, y} = xV y and inf {x, y} = x A y.

Recall that a subset S of a vector space is called circled (or balanced) whenever Ax G S

holds for all x 6 S and all A G R with |A| 1.