# 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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### What is the pushforward measure of the Wiener measure on $C^0([0,1])$ when we apply Legendre Fenchel transform?

Let us consider a Brownian motion on $[0,1]$ denoted by $(W_t)_{t \in [0,1]}$. Let us consider the function $f_W(p) = \sup_{0 \le t \le 1 } (pt - W_t)$ What is the distribution of $f_W(p)$ for a fixed ...
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### Log-sum-exp is the conjugate of relative entropy: a quick proof

Let $(\Omega, \mathcal A, dm)$ be a measure space. For measurable functions $f\colon \Omega\to \mathbb R$ such that $\exp(f)$ is integrable, and for $\rho>0$ such that $\int \rho\, dm=1$, we define ...
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### Why is the Legendre transform (of vector bundles) a smooth morphism $\mathbf FL:E\to E^*$?

The Legendre transform of convex functions $\mathbb R^n\to\mathbb R^n$ can be given a nice geometric interpretation as a way to characterise a function via the set of the tangent spaces to its graph. ...
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### Does the derivative of the Legendre transform satisfy $f^{*\prime}(x)=(f')^{-1}(x)$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth convex function with an invertible first derivative, define the Legendre transform by \begin{eqnarray} f^*(x):=\max\limits_{p \in [a,b]}\{px-f(p)\} ...
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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 ...
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 ...