The Legendre transform is a way to describe a function in terms of it’s supporting hyperplanes. It is most useful for closed convex functions – in that case the supporting hyperplanes retain all the information present in the original function.. Here we motivate the (multidimensional) Legendre transform as an intuitive encoding of these supporting hyperplanes.
See also
http://jmanton.wordpress.com/2010/11/21/introduction-to-the-legendre-transform/
www.mia.uni-saarland.de/Teaching/NAIA07/naia07_h3_slides.pdf
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Thanks for the description. I knew that the Legendre-Fenchel transform has a geometric interpretation, but it’s surprisingly hard to find a short and intuitive statement of what that is.
Yeah, for a long time I had the feeling that there was something geometric going on, but it wasn’t until I read Jonathan Manton’s article (linked above) that it finally made sense.