Definition:Convex Real Function/Definition 1/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:

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


ConvexFunction1.png


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


Also defined as

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

$\forall x, y \in I, x \ne y: \forall t \in \left({0 \,.\,.\, 1}\right) : f \left({t x + \left({1 - t}\right) y}\right) < t f \left({x}\right) + \left({1 - t}\right) f \left({y}\right)$


Also see