Epigraph and hypograph of functions

Definition

Let (S,𝒪)(S, \mathcal{O}) be a topological space and let f:S→ℝ̂f : S \longrightarrow \hat{\mathbb{R}} be a function. Its epigraph is the set epi(f)={(x,y)∈S×ℝ∣f(x)≤y}\mathrm{epi}(f) = \{(x,y) \in S \times \mathbb{R} \mid f(x) \leq y\}

The hypograph of ff is the set hyp(f)={(x,y)∈S×ℝ∣y≤f(x)}\mathrm{hyp}(f) = \{(x,y) \in S \times \mathbb{R} \mid y \leq f(x)\}

References:

  1. https://www.cs.umb.edu/~dsim/cs724/sconvs3.pdf