Isn't Jensen's inequality just the definition of a convex function?
$\phi(\sum_{n}\lambda_{i}x_{i}) \leq \sum_{n}\lambda_{i}\phi(x_{i})$
Looking at the wiki page of Jensens inequality the diagram is just the figure of a convex function (indeed the same figure appears on the page of convex functions).
What exactly is the difference between the case of $n=2$ and arbitrary $n$ when they both state that the function is convex? Were one to suppose that the case of arbitrary $n$ is for the multidimensional case, there doesn't seem to be a multivariate equivalent of $\phi$ since a multivariate function is expected to look like $\Phi(X_{1}, X_{2}, ... X_{n})$ but the $\phi$ in this case always looks like a univariate $\Phi(X)$, where the $X$ can be $\lambda_{1}x_{1}$ or $\lambda_{1}x_{1} + \lambda_{2}x_{2}$ or $\lambda_{1}x_{1} + \lambda_{1}x_{1} + \cdots + \lambda_{n}x_{n}$ (where $\sum_{n}\lambda_{i} = 1$) : ultimately just a single term.
So, going by that Wiki figure (on the pages linked above), if a function is convex between $x_{1}$ and $x_{2}$, is there a chance adding numerous other points in between (which is basically the statement of the $n$ dimensional case) would change that fact (thus needing a proof)?
I did spend a while looking through this other closely related math.SE discussion but that did not answer any of my questions - the final comment there is just that nice little proof by induction, which I am already perfectly fine with. What I don't understand if the generic case says something non-obvious. Please explain if it does.
Apologies if the question is a bit vague. Even though I can do the proofs by induction, I am myself unclear if it signifies anything significant.