Convex Function

A function \(f\) is convex if its values between any two points upon its graph do not lie above the straight line \(g\) connecting those points. Specifically
\[ g(x) = \frac{x-x_0}{x_1-x_0} \times f\left(x_1\right) + \frac{x_1-x}{x_1-x_0} \times f\left(x_0\right)\\ \forall x \in \left[x_0, x_1\right] \quad f(x) \leqslant g(x) \]
where \(\in\) means within, the square brackets represent a closed interval and \(\forall\) is the universal quantifier.

If they lie below that straight line except at those points then it is is known as a strictly convex function.