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I was reading the following

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My question In the first line of the proof, is the author assuming that such $\mathbf{x}$ exists such that $\mathbf{p}=\nabla f (\mathbf{x}(\mathbf{p}))$? To clarify, by definition $$ f^{*}(\mathbf{p})=\sup _{\mathbf{x}} (\mathbf{p}\cdot \mathbf{x}-f(\mathbf{x})) $$ then I would assume that the author means to differentiate what we have in the supremum and claim that at the stationary point we must have the maximum. I would agree with this if we were told that a stationary point exists, however, this is not the case.

Maths Wizzard
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  • This is a consequence of the Legendre-Transform being finite (by definition of the domain) and convex (or rather $-f$ being concave). It must then attain its supremum. – DrShredz Aug 16 '22 at 22:54
  • By the way, this result holds even if $f$ is not differentiable -- it is sufficient to assume that $f$ is convex and closed (that is, lower semicontinuous). Here's how I think about it: https://math.stackexchange.com/a/2226937/40119 – littleO Aug 16 '22 at 23:02
  • Hi @DrShredz, thanks for the comment. I know that the supremum has to be attained, but I am not sure that the supremum has to be a stationary point in the sense given in the first line of the proof. – Maths Wizzard Aug 16 '22 at 23:02

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