§ Mean, Median and Jensen's
The intuition for Jensen's is typically presented as:

 \ /
 \ * /
 \ /
 @

+x>

*
is the average of the $f(x)$ 
@
is the $f$ of average of the x's.  I wish to reinterpret this: the
@
is at the median of the $f(x)$s. So Jensen is maybe saying that the value at the median is lower than the mean of the values in this case due to the convexity of $f$.  In some sense, this tells us that the "data" $\{ f(x): l \leq x \leq r \}$ is skewed in such a way that median is lower than the mean.
 I don't know if this perspective helps, or even if it is correct, but I wish to dwell on this perspective since it's one I don't use often. I've been thinking more along these lines due to competitive programming, and I quite enjoy the change!