Introduction to the Legendre Transform

Jonathan Manton's Blog

I have not come across an introduction to the Legendre transform which was entirely intuitive and satisfying.  The article Making Sense of the Legendre Transform is one such attempt and has a number of attractive features.  However, like other attempts, it immediately links the Legendre transform to re-parametrising a function by its derivative, yet this link is not made crystal clear. Below, I present an alternative introduction to the Legendre transform which takes as its starting point the fact that a convex set is uniquely defined by the collection of its supporting hyperplanes.

My purpose is not to be rigorous, but instead, to present just enough facts for the reader to be comfortable with the basic ideas surrounding the Legendre transform and to tune the reader’s receptiveness to other pedagogic material on Legendre transforms.

The Legendre Transform

Readers are invited to draw their favourite convex function $latex f: mathbb{R} rightarrow…

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