Chapter 3

Convex functions

3.1 Basic properties and examples

3.1.1 Definition

A function f : R^{n} → R is convex if dom f is a convex set and if for all x, y ∈ dom f, and θ with 0 ≤ θ ≤ 1, we have

(3.1) |

Geometrically, this inequality means that the line segment between (x, f(x)) and (y, f(y)), which is the chord from x to y, lies above the graph of f (figure 3.1). A function f is strictly convex if strict inequality holds in (3.1) whenever x y and 0 θ 1. We say f is concave if −f is convex, and strictly concave if −f is strictly convex. ...

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