Iterative Optimization in Inverse Problems

80 Iterative Optimization in Inverse Problems

Definition 6.6 The convex hull of a set S, denoted conv(S), is the small-

est convex set containing S, by which we mean that if K is any convex set

containing S,thenK must also contain conv(S).

One weakness of this definition is that it does not tell us explicitly what the

members of conv(S) look like, nor precisely how the individual members

of conv(S) are related to the members of S itself. In fact, it is not obvious

that a smallest such set exists at all. The following proposition remedies

this; the reader is asked to supply a proof in Exercise 6.2 later.

Proposition 6.1 The convex hull of a set S is the ...