v2009.01.01 - Convex Optimization

34 CHAPTER 2. CONVEX GEOMETRY
2.1 Convex set
A set C is convex iff for all Y, Z ∈ C and 0≤µ≤1
µY + (1 − µ)Z ∈ C (1)
Under that defining condition on µ , the linear sum in (1) is called a convex
combination of Y and Z . If Y and Z are points in real finite-dimensional
Euclidean vector space R n or R m×n (matrices), then (1) represents the
closed line segment joining them. All line segments are thereby convex sets.
Apparent from the definition, a convex set is a connected set. [222,3.4,3.5]
[37, p.2] A convex set can, but does not necessarily, contain the origin.
An ellipsoid centered at x = a (Figure 12, p.38), given matrix C ∈ R m×n
{x∈ R n | ‖C(x − a)‖ 2 = (x − a) T C T C(x − a) ≤ 1} (2)
is a good icon for a convex set.
2.1.1 subspace
A nonempty subset of real Euclidean vector space R n is called a subspace
(formally defined in2.5) if every vector 2.1 of the form αx + βy , for
α,β∈R , is in the subset whenever vectors x and y are. [215,2.3] A
subspace is a convex set containing the origin 0, by definition. [266, p.4]
Any subspace is therefore open in the sense that it contains no boundary,
but closed in the sense [222,2]
It is not difficult to show
R n + R n = R n (3)
R n = −R n (4)
as is true for any subspace R , because x∈ R n ⇔ −x∈ R n .
The intersection of an arbitrary collection of subspaces remains a
subspace. Any subspace not constituting the entire ambient vector space R n
is a proper subspace; e.g., 2.2 any line through the origin in two-dimensional
Euclidean space R 2 . The vector space R n is itself a conventional subspace,
inclusively, [197,2.1] although not proper.
2.1 A vector is assumed, throughout, to be a column vector.
2.2 We substitute the abbreviation e.g. in place of the Latin exempli gratia.