The epigraph of a function $f:\mathbb{R}^{n}\to [-\infty,+\infty]$ is the set of points $(x,\mu)\in\mathbb{R}^{n+1}$ satisfying $f(x)\leq \mu$, and is denoted by $\mathrm{epi}(f)$. I somehow got convinced that any such function can be recovered from its epigraph via the formula $f(x)=\inf\{\mu|(x,\mu)\in \mathrm{epi}(f)\}$ (mainly because of its geometric intuition), but now I have second thoughts. Is this formula true?
$\endgroup$ 2 Reset to defaulta function and its epigraph
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