I am reading through Jensen’s inequality and its proofs. I want to find an alternate proof to the tangent line proof.
My attempt:
If $f(x)$ is a convex function, then $\{x_{i},f(x_{i})\}\ ,\ i=1,2,...,n$ are vertices of a convex $n$ - gon, the interior of which is never below $f(x)$.
For $a_{i}\geq 0\ , \ \sum_{i=1}^{n}{a_{i}}=1$, the $n$ - gon’s interior point $\left\{\sum_{i=1}^{n}{a_{i}x_{i}}\ , \sum_{i=1}^{n}{a_{i}f(x_{i})}\right\}$ is never below $\left\{\sum_{i=1}^{n}{a_{i}x_{i}}\ ,f\left(\sum_{i=1}^{n}{a_{i}x_{i}}\right)\right\}$.
This is equivalent to $\sum_{i=1}^{n}{a_{i}f(x_{i})}\geq f\left(\sum_{i=1}^{n}{a_{i}x_{i}}\right)$
Is this a valid proof?