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
(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.
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} .
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.
Previous Page Next Page