I am trying to build intuition on Legendre transforms. Arnold's Mathematical Methods of Classical Mechanics has some nice geometric interpretations, but he does not provide a proof that the Legendre transform of a convex function is convex. I know we can prove it by applying some inequalities, but is there a nice geometric argument (with a picture) that allows us to see that it is true?
The proof you seek, and the geometric insights you desire, and the links to physical dynamics, all are given in an outstanding survey article:
For the convexity proof in particular, see Equation 47 and the discussion that follows.