Weird fact of the day: Suppose , then:
What does it really mean? It means: if you take some positive numbers , raise each to the power of n, add them all together, take the n-th root of the sum, then if you start to increase n the result will start to approach the greatest of the numbers you started with.
Why is that weird? Because the procedure is symmetric with respect to all the numbers , and the result depends only on one of them, the greatest one (Let’s call it ). All the other k-1 numbers are pretty much completely irrelevant. You can set them all to zero and it will not change the outcome, because all the work is done by , and by alone. How does the procedure “determine” which number is the greatest?
How do we prove this amazing fact? Like this:
We already defined . Now:
, therefore , the end.
Does it work with other families of functions or only with powers and roots? The proof only uses some monotonicity and the fact that , so it should also work, for example, when , meaning that:
I can’t think of a good way to completely characterize the class of functions, for which the proof holds. Can you?
What does this mean geometrically? It means that as n increases, level surfaces of a function defined in a k-dimensional space, will start to resemble a k-dimensional cube, because a cube is a level surface of a function . You might remember that we already spoke about this in my previous post titled “On Norms”.
Have you made a cool-looking but completely pointless animation to demonstrate this phenomenon? I thought you’d never ask:
(you may or may not need to click on it)
Does this fact have any useful applications? Not that I know of, which proves that this is some high quality math, and not same applied rubbish. (Just kidding, applied math can be beautiful too. Once in a thousand years.)