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Halfspaces are Convex Sets

Last updated Nov 1, 2022

# Statement 1

Let $V$ be an Inner Product Space over $\mathbb{R}$, let $a \in V$, $b \in \mathbb{R}$, and let $H$ be the Closed Halfspace $$H = {x : \langle x, a \rangle \geq b}.$$ Then $H$ is a Convex Set.

# Proof

TODO This should follow from the fact that Inner Products are Linear in their first argument

# Statement 2

Let $V$ be an Inner Product Space over $\mathbb{R}$, let $a \in V$, $b \in \mathbb{R}$, and let $H$ be the Open Halfspace $$H = {x : \langle x, a \rangle > b}.$$ Then $H$ is a Convex Set.

# Proof

TODO follows from statement 1 and Interior of Convex Set is Convex.