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1. Cones

decomposition and interpolation properties play an important role and will

be featured frequently throughout the book.

1.1. Wedges and cones

In this section we shall present some basic properties of wedges and cones

in vector spaces. We start by recalling that a subset C of a vector space is

called convex if x, y e C and 0 A 1 imply Xx + (1 — X)y G C.

Definition 1.1. A nonempty subset W of a vector space is said to be a

wedge if it satisfies the following two properties:

(1) W + W C

W,1

(2) aW C W for all a 0.

Clearly, every wedge is a convex set. Moreover, if W is a wedge, then

it is not difficult to see that the set W fl (—W) is a vector subspace. When

this vector subspace is trivial, the wedge is called a cone. Precisely, we have

the following definition.

Definition 1.2. A nonempty subset K of a vector

space is said to be a cone if it satisfies the following

three properties:

(1) /C + /CC/C

(2) a/CC/C for all a 0,

(3) /Cn(-/C ) = {0} .

A cone with vertex x is any set of the form x + K,

where K is a cone?

Obviously, every cone is a wedge but a wedge need not be a cone. For in-

stance, every vector subspace is a wedge but only the trivial vector subspace

is a cone. Clearly, every cone (being a wedge) is automatically a convex set.

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Throughout this monograph we shall use the following standard notation. If A and B are

subsets of a vector space X and a, (3 € M, then we let

a A + (3B = {x E X: 3 a € A and b € B such that x = aa + (3b} .

o

Many authors define a cone to be simply a nonempty subset /C of a vector space such that

ax E K holds for all x G /C and all a 0. In the literature a wedge is quite often referred to as a

convex cone. A cone as defined in our Definition 1.2 is also known as a pointed convex cone

or as a pointed convex cone with vertex at zeroor even as a cone with vertex at zero.