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Do Legendre transforms have any applications that do not appeal to convexity?

What is the intuitive interpretation of the Legendre transform of a non-convex function?

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I can answer your second question.

Recall that by convex duality, applying the Legendre transform twice on a convex (and coercive) function gives you back the original function. Now even if you plug in a non-convex function $W$ instead, the Legendre transform $W^*$ will be convex and coercive (one can see that by the definition) and so is $V:=(W^*)^*$. So clearly the convex duality will not carry over to the non-convex case. However one can show (under appropriate assumptions of course) that $V$ is the convexification or convex hull of $W$, that is roughly (here $W:\mathbb{R}\rightarrow\mathbb{R}$) the function whose graph is the boundary curve of the convex hull of the set $\{(x,y)\in\mathbb{R}^2\,|\,y\ge f(x)\}$. By convex duality we conclude that $V^*=((W^*)^*)^*=W^*$, so the Legendre transform of a non-convex function is the Legendre transform of it's convexification.

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Thanks! Is it possible to give me some references that would allow me to study these points in greater detail? –  Per Oct 14 '12 at 10:02
    
E.g. Evans' book "Partial Differential Equations" contains a proof of convex duality (in a chapter on Hamilton-Jacobi equations I think). For the non-convex case, I think looking at $(W^*)^*$ for the example $W(p)=(1-p^2)^2$ is instructive (Hint: don't try calculating $W^*$ first, it's a mess and doesn't help). –  Your Ad Here Oct 14 '12 at 11:47
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I would like to propose a different answer to the second part of your question. Let us examine a simple example of a non-convex function, $A(x) = x^2-x^4$, which of course looks like this:

enter image description here

(plots are from Wolfram|Alpha.)

Let us define $y=\frac{dA}{dx}=2x-4x^3$ and consider the Legendre transform $B = xy-A(x) = x^2 - 3x^4$. Solving explicitly to get $B$ as a function of $y$ involves solving a cubic equation and doesn't seem to lead anywhere nice, but we can easily do a parametric plot of $B(x)$ against $y(x)$ to see what it would look like. The result is this: ($B$ is the vertical axis and $y$ is the horizontal axis)

enter image description here

I believe that Legendre transforms of functions with a single non-convex region will always be qualitatively similar to this. In general the Legendre transform of such a function will be a multiple-valued function with this distinctive "swallowtail" shape. It won't always be symmetrical of course, but there will always be two points where the curve reverses direction and one point where it crosses itself. The direction of curvature of the three line segments will also always follow the pattern of this plot.

The reason the curve reverses direction is simply that $y$ is the slope of the $A(x)$ curve. For a convex function, $\frac{dy}{dx} = \frac{d^2 A}{dx^2}$ is always negative, so $y$ changes monotonically with $x$; but for a non-convex function there is a region where $\frac{dy}{dx} = \frac{d^2 A}{dx^2}>0$, corresponding to a reversal in the slope of $y(x)$. Indeed we can see this non-monotonicity by plotting $y=2x-4x^3$ against $x$:

enter image description here

If there are multiple concave regions then there will be multiple reversals of the sign of $dy/dx$, corresponding to multiple swallowtails in the Legendre transform. I think these multiple swallowtails can be sort of nested inside one another.

If one replaces the non-convex function with its convex hull then the "swallowtail" will disappear, with the point where the curve crosses itself becoming a "kink" in the curve, where the second derivative becomes infinite. This results in a single-valued convex function.

As for whether this interpretation is useful, I'm not sure but I think so. It seems related to the swallowtail catastrophe (although I don't know much about catastrophe theory), and it also seems related to first-order phase transitions in statistical mechanics. These can be seen as being due to a non-convexity in the entropy (which then gets replaced by its convex hull). This is equivalent to a swallowtail in the free energy being replaced by a kink, since the two are Legendre transforms of one another. I've never seen this swallowtail curve depicted in a thermodynamics text, but it's not uncommon to see plots corresponding to the $y(x)$ plot above.

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