Epigraph

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Definition

Let $S\in\mathcal{P}(\mathbb{R}^n)$ be an arbitrary subset of $\mathbb{R}^n$ and let $f:S\rightarrow\mathbb{R}$ be a function, the epigraph of $f$, $G\subseteq\mathcal{P}(\mathbb{R}^{n+1})$ is defined as followsTemplate:RAFCIRAMP:

• $G:\eq\{ (x_1,\ldots,x_n,y)\in\mathbb{R}^{n+1}\ \vert\ x\in S\wedge y\eq f(x)\}$

See also

• A function is convex if and only if its domain is convex and its epigraph are convex sets