Convex sets and convex functions

Many important inequalities depend upon convexity. In this chapter, we shall establish Jensen's inequality, the most fundamental of these inequalities, in various forms.

A subset C of a real or complex vector space E is convex if whenever x and y are in C and 0 ≤ θ ≤ 1 then (1 − θ)x + θy ∈ C. This says that the real line segment [x, y] is contained in C. Convexity is a real property: in the complex case, we are restricting attention to the underlying real space. Convexity is an affine property, but we shall restrict our attention to vector spaces rather than to affine spaces.

Proposition 4.1.1A subset C of a vector space E is convex if and only if whenever x 1, …, xn ∈ C and p 1, …, pn are positive numbers with p 1 + … + pn = 1 then p 1 x 1 + … + pnxn ∈ C.

Proof The condition is certainly sufficient. We prove necessity by induction on n. The result is trivially true when n = 1, and is true for n = 2, as this reduces to the definition of convexity. Suppose that the result is true for n − 1, and that x 1, …, xn and p 1, …, pn are as above.