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!