Jensen's Inequality - A Visual Intuition

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2024-09-16 19:30:06

Jensen’s inequality finds widespread application in mathematical proofs. I am fond of a particular intuitive explanation of it, which doesn’t seem to be very popular. I will try to present it in brief here.

I am not sure when this argument originated, but Google does turn up a paper (Needham, 1993). Even if this is not the source, it is a good reference. On a related note, the author of the paper, Tristan Needham, has the very well-reviewed book “Visual Complex Analysis” to his credit.

For our purposes, we can think of the center of mass of an object as that one point where you could hold up an object. Easy to intuit and easily shown with pictures. Below, you see the objects - a toy bird and a baseball bat - held up at their CM.

We wouldn’t be looking at the CMs of bodies like the toy bird or a baseball bat, but of groups of disconnected masses, like in the figure on the left below; here, a mass is represented by a circle, where its size corresponds to its mass. The CM is shown with a red square. If the disconnected-ness of the masses seems confusing, you could assume that the masses are affixed to a Plexiglas sheet of negligible weight, as shown on the right. Again, the intuition for the CM is that you can hold up this system at its location.

The key takeaway - which might seem silly because it’s obvious to our physical intuition - should be that the CM always within the convex hull of the masses. There are great descriptions of convex hull online of course, but a description I like is: Think of the given points as locations of nails driven into a board. If you now were to snap a rubber band around these nails, the shape it ends up with is the convex hull. Let’s look at the masses above again - now with the convex hull also shown.