Hypograph

In mathematics , the hypograph of a real-valued function denotes the set of all points that are on or below its graph . ${\displaystyle f}$

definition

Be . The function's hypograph is defined by ${\displaystyle X\subset \mathbb {R} ^{n}}$ ${\displaystyle f\colon X\to \mathbb {R} }$

${\displaystyle \mathrm {hypo} \,f:=\left\{(x,\mu )\in X\times \mathbb {R} \,:\,\mu \leq f(x)\right\}\subseteq X\times \mathbb {R} \,.}$

If the image space of the function is provided with a generalized inequality , then the hypograph is defined as ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \preccurlyeq _{K}}$

${\displaystyle \mathrm {hypo} \,f:=\left\{(x,\mu )\in X\times \mathbb {R} ^{n}\,:\,\mu \preccurlyeq _{K}f(x)\right\}\subseteq X\times \mathbb {R} ^{n}}$.

properties

Be . The following applies to functions : ${\displaystyle X\subset \mathbb {R} ^{n}}$ ${\displaystyle f\colon X\rightarrow \mathbb {R} }$

• ${\displaystyle f}$is concave if and only if the hypograph of forms a convex set. ${\displaystyle f}$
• ${\displaystyle f}$is above- half continuous if and only if the hypograph of forms a closed set. ${\displaystyle f}$
• If an affine-linear function, then its hypograph defines a half-space in . ${\displaystyle f}$ ${\displaystyle X}$

See also

• Epigraph (mathematics)

Individual evidence

1. ^ Wilhelm Rödder, Peter Zörnig: Business Mathematics for Study and Practice 3 - Analysis II . Springer, 1997, ISBN 978-3-540-61716-7 , pp. 55 .