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?
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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.