# Is there a geometric argument that the Legendre transform of a convex function is convex?

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?

## 2 Answers

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.

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).