PomeDefinition 21.11.22.2E5E29E9A58B Let $F\in \mathrm{Conv}(N)$ be a convex function. We define the Legendre transform of $F$ as the function $F^*\in \mathrm{Conv}(M)$:
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21.11.23
\begin{equation} F^*(m)\coloneqq \sup _{n\in N} \left( \langle m,n \rangle -F(n) \right)=\sup _{n\in \operatorname{RelInt}\operatorname{Dom}F} \left( \langle m,n \rangle -F(n) \right). \end{equation}
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