Definition:Convex Real Function/Definition 2

Definition

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

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

$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$

Hence a geometrical interpretation: the slope of $P_1 P_2$ is less than or equal to that of $P_2 P_3$:

Strictly Convex

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

$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} < \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$

Geometric Interpretation

Let $f$ be a convex real function.

Then:

for every pair of points $A$ and $B$ on the graph of $f$, the line segment $AB$ lies entirely above the graph.

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): $\S 12.13$