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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:

@article{Author = {Zia, R. K. P. and Redish, Edward F. and McKay, Susan R.}, 
Journal = {American Journal of Physics}, 
Number = {7}, Pages = {614-622}, 
Title = {Making sense of the {L}egendre transform}, 
Volume = {77}, Year = {2009}}

For the convexity proof in particular, see Equation 47 and the discussion that follows.

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If you draw a collection of straight lines on a paper and you inspect the "upper contour" of these lines, you notice that this contour is convex. Similarly, the Legendre transform of a function is basically a supremum (=upper contour) over a family of affine functions (=straight lines).

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