Here I give a visual proof of Jensen's Inequality, using convex
analysis.
Shown below is a convex function and the convex hull of its plot.
A convex function defined on domain X (red), and the hull
of its plot (gray).
The plot of the function f(x) is shown in red. It is
defined on domain X. The expression f(X)
denotes the image of f in the same order as elements in
X. The expression \langle w, X \rangle is
shorthand for the inner product \int_X{dx\, w(x) x} etc.
The hull is shown in gray. In the above graph, we show an arbitrary hull
point, and the plot point directly below it. By "directly below", I mean the
x-coordinate is the same, and the y-coordinate is less than or equal.
Two things are guaranteed. First, by definition of hull, there is
a hull point (\langle w, X \rangle, \langle w, f(X) \rangle)
for every possible choice of convex combination w(x).
Second, by definition of convex function, every hull point has a plot
point on or directly below it.
Considering these two points, we have
\begin{aligned}
x &= \langle w, X \rangle && \text{x is directly...} \\
f(x) &\le \langle w, f(X) \rangle && \text{...below the hull point for this w}\\
f(\langle w, X \rangle) &\le \langle w, f(X) \rangle && \text{Jensen's Inequality}
\end{aligned}
which is true for any choice of w. The final result is
Jensen's Inequality. It says that a convex combination w of
the function values (in domain order) will be not less than the function
value of that same convex combination of the points in the domain.