The proof of Young’s inequality is pretty neat but has the „magically think of taking a log of an arbitrary expression which happens to work” step. But it clarifies why the reciprocals of exponents have to sum up to 1: they are interpreted as probabilities when calculating expected value.
Here’s how I like to conceptualise it: bounding mixed variable product by sum of single variable terms is useful. Logarithms change multiplication to addition. Jensen’s inequality lifts addition from the argument of a convex function outside. Compose.
You've got a product on one side and what looks like a convex combination on the other, taking the log and applying Jensen's inequality isn't as big a leap as it may sound.
Agreed, provided you have both sides of the inequality. Coming up with that particular convex combination is a bit of a leap that’s not super intuitive to me.
if you work with a lot of convex optimization, it comes up pretty often. for example, if you learn fenchel conjugates, the lead up and motivation to learning them will often necessitate proving young's inequality with jensen's inequality. that is why learning different maths is cool. you intuit some ways to reshape the problem in order to make these "not super intuitive" connections.
It often happens that coming up with the right theorem is a lot harder than finding its proof, but that's life. You can't have everything be easy, otherwise we'd have finished by now.
Here’s how I like to conceptualise it: bounding mixed variable product by sum of single variable terms is useful. Logarithms change multiplication to addition. Jensen’s inequality lifts addition from the argument of a convex function outside. Compose.