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Open Halfspace

Last updated Nov 1, 2022

# Definition

Let $V$ be an Inner Product Space

over

\mathbb{R}

. An Open Halfspace is a Set of the form

H = {x : \langle x, a \rangle > b}

for any

a \in V

and

b \in \mathbb{R}

.

# Remarks

An Open Halfspace

Open Halfspace

Definition Let $V$ be an over $\mathbb{R}$ . An is a of the form $$H = \{x : \langle x,...

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is Open

Open

Definition Suppose $(X, \tau)$ is a . Then $U \subset X$ is if $U \in \tau$ ....

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. TODO

TODO

...

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- prove this.
If $H$ is an Open Halfspace

Open Halfspace

Definition Let $V$ be an over $\mathbb{R}$ . An is a of the form $$H = \{x : \langle x,...

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, then $\text{cl} H$ is a Closed Halfspace

Closed Halfspace

Definition Let $V$ be an over $\mathbb{R}$ . A is a of the form $$H = \{x : \langle x,...

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- TODO

TODO

...

11/7/2022

prove this.
The Boundary

Boundary

Definition Let $X$ be a and let $S \subset X$ . Then the of $S$ , denoted $\text{bd}S$ , is the $$\text{bd}S...

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of a Closed Halfspace

Closed Halfspace

Definition Let $V$ be an over $\mathbb{R}$ . A is a of the form $$H = \{x : \langle x,...

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is a Hyperplane

Hyperplane

Definition Let $V$ be an over $\mathbb{R}$ . A is a of the form $$H = \{x : \langle x,...

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- TODO

TODO

...

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prove this.

# Other Outlinks

Real Numbers

Real Numbers

Definition 1 The unique Complet totally ordere . It is usually denoted $\mathbb{R}$ and is used to represent the continuum. Remarks One...

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Closure

Closure

Definition Let $X, \tau$ be a and let $S \subset X$ . Then the of $S$ , denoted $\text{cl}S$ is $$\text{cl}S :=...

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Backlinks

Closed Halfspace

Closed Halfspace

...\mathbb{R}$. Remarks A is . - prove this. If $H$ is an , then $\text{int} H$ is an - prove this. The of a is a - prove this. Other Outlinks ......

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Halfspaces are Convex Sets

Halfspaces are Convex Sets

...be an over $\mathbb{R}$ , let $a \in V$ , $b \in \mathbb{R}$ , and let $H$ be the $H = \{x : \langle x, a \rangle > b\}.$ Then $H$ is a . Proof follows from statement 1 and......

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Open Halfspace

Open Halfspace

...\mathbb{R}$. Remarks An is . - prove this. If $H$ is an , then $\text{cl} H$ is a - prove this. The of a is a - prove this. Other Outlinks ......

11/7/2022
Optimal Solution of Bounded Linear Program is at Vertex

Optimal Solution of Bounded Linear Program is at Vertex

...P$ of the $P$ defined by $A \mathbf{x} \leq \mathbf{b}$ s.t. $\max \mathbf{c}^{T} \mathbf{v}^{} = \max \mathbf{c}^{T} \mathbf{x}^{}$ . That is, $\mathbf{v}^{*}$ is an optimal solution. Proof - this requires convexit of the and s. Alternatively, could use......

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