prove modified jensen's inequality

Question

My question is I'm trying to prove modified jensen's inequality so given a convex function f and $$ E= \{\sum_{0}^{n} \lambda_{i}x_{i} | \lambda_{i}>=0,\sum_{0}^{n} \lambda_{i}=1 \} $$

I want to prove that for all x belonging to E, $f(x) <= max_{i\in\{0...n\} } f(x_{i})$

I assume I should be using the original jensen's inequality but I can't seem to find a solution

Answer 1

Assuming $f$ is convex, this is immediate from the standard Jensen inequality, because a weighted average of the numbers $f(x_i)$ is at most their maximum.

Without assuming convexity of $f$, the inequality may fail.