Definition:Convex Real Function/Definition 3/Strictly

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Definition

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


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

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


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


ConvexFunction3.png


Also see