ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Binary endorelation
Preordering relation
Partial ordering relation
Partially ordered set
Maximal element
Maximum element
Set upper bound
Set of upper bounds
Supremum element
Map supremum element
Definition D5220
Convex conjugate function
Formulation 2
Let $f : X \to \mathbb{R}$ be a D4364: Real function such that
(i) $X \subseteq \mathbb{R}^N$ is a D78: Subset of $\mathbb{R}^N$
(ii) \begin{equation} Y : = \left\{ y \in \mathbb{R}^N : \sup_{x \in X} (y^T x - f(x)) < \infty \right\} \end{equation}
The convex conjugate of $f$ is the D4364: Real function \begin{equation} Y \to \mathbb{R}, \quad y \mapsto \sup_{x \in \mathbb{R}^N} (y^T x - f(x)) \end{equation}