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

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.