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

Greetings I found your website by mistake when i searched Google for this concern, I have to point out your page is really very helpful I also really like the design, it is wonderful!

It is very helpful to me! Very nice!

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.

f^* is convex so I’m not sure if the graph for f^* is right. I think it should be upside down!

Yes of course, it was upside down and is corrected now.

Very nice explanation! Thanks a lot.

Joaquin Delgado

Thanks, I haven’t heard about this interpretation before. Really helpful

You sir are a gentleman and a scholar.

Thanks a lot, the plots really help here ! The idea just looks so beautiful now