A half-space in $\mathbb{R}^{k}$ is a set of the form $$H(\mathbf{u},\beta)=\{ \mathbf{x}\in\mathbb{R}^{k}:\mathbf{u}^{T}\mathbf{x}+\beta\ge 0 \}$$where $\mathbf{u}\in\mathbb{R}^{k} \ (\mathbf{u}\not=0)$ and $\beta\in\mathbb{R}$.

A hyperplane $\Lambda(\mathbf{u},\beta)$ is the boundary of the $H(\mathbf{u},\beta)$ i.e. $$\Lambda(\mathbf{u},\beta)=\{ \mathbf{x}:\mathbf{u}^{T}\mathbf{x}+\beta=0 \}$$ ## Intuition The this is the divider for the half-space

A set $R\subset\mathbb{R}^{k}$ is a convex polytope if it can be written as a finite intersection of s i.e. $$R=\bigcap_{i=1}^{L}H(\mathbf{u}_{i},\beta_{i})$$for some $\mathbf{u}_{1},\dots,\mathbf{u}_{L}\in\mathbb{R}^{k}$ and $\beta_{1},\dots,\beta_{L}\in\mathbb{R}$. ## Intuition Everything within the convex polytope and on the boundary is inside the space. Hence this convex polytope forms an additive group.