The Theorem of the Day: Jensen’s Inequality

If $f$ is a real convex function, $x_1,...,x_n$ are values in the domain of $f$, and $a_1,...,a_n$ are positive real numbers summing to $1$ then

$f(\sum a_i{x}_i)\le \sum a_{i}f(x_i)$.

The function plotted here, showing relationship of radius in cm (horizontal axis) to surface area in $c{m}^2$ (vertical axis) is a convex function: for any two points on the plotted curve, all internal points on the straight line joining them lie above the curve.

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