Jordan–Wigner transformation

In mathematics, the hypograph or subgraph of a function f : Rⁿ → R is the set of points lying on or below its graph:

$\text{hyp}f=\{(x,\mu ):x\in {\mathbb{ℝ}}^n,\mu \in \mathbb{ℝ},\mu \le f(x)\}\subseteq {\mathbb{ℝ}}^{n+1}$

and the strict hypograph of the function is:

${\text{hyp}}_{S}f=\{(x,\mu ):x\in {\mathbb{ℝ}}^n,\mu \in \mathbb{ℝ},\mu <f(x)\}\subseteq {\mathbb{ℝ}}^{n+1}.$

The set is empty if $f\equiv -\mathrm{\infty }$ .

Similarly, the set of points on or above the function's graph is its epigraph.

Properties

A function is concave if and only if its hypograph is a convex set. The hypograph of a real affine function g : Rⁿ → R is a halfspace in Rⁿ⁺¹.

A function is upper semicontinuous if and only if its hypograph is closed.