Supporting hyperplane

Builds on Euclidean space Dot product Boundary Support function

Definition

Let $n \ge 1$ and let $S \subseteq \R^n$ . A hyperplane $H = \set{x \in \R^n : a \cdot x = b}$ , with $a \in \R^n \setminus \set{0}$ and $b \in \R$ , is a supporting hyperplane of $S$ if:

$S$ lies in one of the closed half-spaces of $H$ , say $a \cdot x \le b$ for all $x \in S$ ;
$H$ contains at least one point of the boundary of $S$ .

Here $a \cdot x$ is the dot product. If $x_0 \in H$ is such a boundary point, $H$ supports $S$ at $x_0$ .

Remarks

Example: for the closed unit disc $D \subseteq \R^2$ and $a = (0, 1)$ , $b = 1$ , the line $y = 1$ supports $D$ at $(0, 1)$ . Each $(x, y) \in D$ has $y \le 1$ , and $(0, 1)$ is a boundary point of $D$ on the line.

Non-example: the line $y = 2$ satisfies clause 1 for $D$ , but it misses $D$ and its boundary, so clause 2 fails. The line $y = 0$ meets the boundary of $D$ but has points of $D$ on both sides, so clause 1 fails.

In terms of the support function $h_S$ , when $S$ is closed and bounded and nonempty, the hyperplane $a \cdot x = h_S(a)$ is the supporting hyperplane with outer normal $a$ .

Reading path

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Boundary concept
Dot product concept
Euclidean space object
Support function concept
Supporting hyperplane definition

Tag	`XVPJ44E0KW`
URI	`m2://XVPJ44E0KW`
Kind	definition
Writing	W0 stub
MSC	52A20
Defines	Supporting hyperplane
Uses	Euclidean space, Dot product, Boundary, Support function

Supporting hyperplane

Definition

Remarks

Reading path

Further reading

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