Fenchel-Young inequality

Fenchel-Young inequality: Let $f:\mathbb{R}^n\to(-\infty,+\infty]$ be a proper function , and let $f^*$ be its Fenchel conjugate . Then for all $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$,

If $f$ is convex, equality holds if and only if $y\in\partial f(x)$ (equivalently, $x\in\partial f^*(y)$), where $\partial f$ denotes the subdifferential .

This inequality is the basic mechanism behind weak duality in convex primal-dual pairs : conjugate-based dual objectives arise by repeatedly applying Fenchel-Young.