ABSTRACT
In this chapter we introduce convex sets and their representations, properties,
illustrative examples, convexity preserving operations, and geometry of convex
sets which have proven very useful in signal processing applications such as
hyperspectral and biomedical image analysis. Then we introduce proper cones
(convex cones), dual norms and dual cones, generalized inequalities, and sepa-
rating and supporting hyperplanes. All the materials on convex sets introduced
in this chapter are essential to convex functions, convex problems, and duality to
be introduced in the ensuing chapters. From this chapter on, for simplicity, we
may use x to denote a vector in Rn and x1, . . . , xn for its components without explicitly mentioning x ∈ Rn.
