Obtained from https://inst.eecs.berkeley.edu/~ee127/sp21/livebook/def_epi.html

Function $f:\R^n \rightarrow \R$ and has 4 relevant sets:

• graph and epigraphs (subsets of $\R^{n+1}$)

• level and sub-level sets (subsets of $\R$)

• graph: set of input-output pairs that $f$ can attain (line in below graph)

$$ G(f) \coloneqq \{(x, f(x)) \in \R^{n+1} : x \in \R^n \} $$

• epigraph: describes set of input-output pairs $f$ can achieve and additionally includes the region with higher values (line and blue region in below graph)

$$ \mathbf{epi} f \coloneqq \{ (x, t) \in \R^{n+1} : x \in \R^n, t \geq f(x) \} $$

Graph (blue line) and epigraph (blue line and light blue region) of function with 1-dimensional mapping) over a domain of -1 to 1

• level sets: set of points that achieve exactly a certain value for $f$

  • $t$-level set of $f$:

    $$ L_t(f) \coloneqq \{ x \in \R^n : x \in \R^n, t = f(x) \} $$

  • 2-dimensions ($n=2$)? level sets are called level curves

• sub-level sets: set of points that achieve at most a certain value for $f$ (includes lower values)

  • $t$-sub-level set of $f$:

  $$ S_t(f) \coloneqq \{ x \in \R^n : x \in \R^n, t \geq f(x) \} $$