Definition:Convex Real Function/Definition 1

Definition

Let $f$ be a real function which is defined on a real interval $I$.

$f$ is convex on $I$ if and only if:

$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$

The geometric interpretation is that any point on the chord drawn on the graph of any convex function always lies on or above the graph.

Strictly Convex

$f$ is strictly convex on $I$ if and only if:

$\forall x, y \in I, x \ne y: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} < \alpha \map f x + \beta \map f y$

Also presented as

By setting $\alpha = \lambda$ and $\beta = 1 - \lambda$, this can also be written as:

$\forall x, y \in I, x \ne y: \forall \lambda \in \openint 0 1: \map f {\lambda x + \paren {1 - \lambda} y} \le t\lambda \map f x + \paren {1 - \lambda} \map f y$

Also known as

A convex function can also be referred to as:

a concave up function

a convex down function.

Also see

• Equivalence of Definitions of Convex Real Function

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 12.13$

{{SourceReview|{{WRT} Also presented as}}

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 12$