It’s neat! To be fair, as a physicist, I did not understand the Legrendre transform essentially until taking convex optimization (where it is known as the Fench

It’s neat! To be fair, as a physicist, I did not understand the Legrendre transform essentially until taking convex optimization (where it is known as the Fenchel conjugate). Many sources, but all of them are reasonable and give a constructive definition that actually explains what it does: we can characterize a function either by its graph, or its supporting hyperplanes (when it is a closed, convex function). While the observation is almost silly, it has very deep consequences for different characterizations of problems and other constructions!