Reciprocal Convexity to reverse the Jensen Inequality
Jensen's inequality is a powerful tool often used in mathematical derivations and analyses. It states that for a convex function $f(x)$ and an arbitrary random variable $X$ we have the following upper bound: $$ f\left(\E X\right) \le \E f\left(X\right) $$
However, oftentimes we want the inequality to work in the other direction, to give a lower bound. In this post I'll outline one possible approach to this.