Questions tagged [legendre-transformation]

For questions about the Legendre transformation, an involution transform commonly used in classical mechanics and thermodynamics as well as for it's generalization, the Legendre–Fenchel Transformation.

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Integral kernel of the Legendre transform

First of all, I'm not sure, but I think the Legendre transform can be seen as a linear operator between the functions on a normed space and the functions on its dual. (A functional analysis approach ...
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31 views

Linking push-forward of measures and the Legendre transform

Suppose that $\rho_1$ and $\rho_2$ are absolutely continuous w.r.t Lebesgue measure on $\mathbb{R}^n$, for which the second moment of both measures are finite. By Brenier's theorem, there exists a ...
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Legendre transform of a convex downward function

I have the following function: $$\lim _{t \rightarrow \infty}-\frac{1}{t} \log \left\langle e^{-\lambda W(t)}\right\rangle=e(\lambda),$$ which is convex downward, where $\langle\exp [-\lambda W(t)]\...
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35 views

Convex conjugate of the exponential, via subdifferentials

Can anyone explain how to go about finding the convex conjugate of $\mathbb{R}\ni x \mapsto e^x$ via the subdifferential convex analysis tricks?
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How can I conduct the Legendre transformation of this problem?

Consider the following function $$f(x) = \left\{\begin{array}-(a^2 - x^2)^{1/2} & \text{if }|x|\leq a \\+ \infty &\text{otherwise}\end{array}\right.$$ and compute its Legendre transform. I ...
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37 views

Legendre transform of a scalar function - getting stuck at inverting vectors

I'm interested in finding the Legendre transform of the following function $$ f(x) = - \log(\langle a,x\rangle) $$ where $a$ and $x$ are both $n$-dimensional (column) vectors. The values of $x$ and $...
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49 views

Uniqueness of the Legendre-Fenchel Transformation

What is the relation between two different functions, say $g(x)$ and $f(x)$, which have the same Legendre–Fenchel transformation $h(s)$? \begin{equation} h(s) = \sup_{x\in I}\{sx - f(x)\} \quad \...