prove modified jensen's inequality 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

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.
