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.