$$ \newcommand{\qed}{\tag*{$\square$}} \newcommand{\span}{\operatorname{span}} \newcommand{\dim}{\operatorname{dim}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\norm}[1]{\|#1\|} \newcommand{\grad}{\nabla} \newcommand{\prox}[1]{\operatorname{prox}_{#1}} \newcommand{\inner}[2]{\langle{#1}, {#2}\rangle} \newcommand{\mat}[1]{\mathcal{M}[#1]} \newcommand{\null}[1]{\operatorname{null} \left(#1\right)} \newcommand{\range}[1]{\operatorname{range} \left(#1\right)} \newcommand{\rowvec}[1]{\begin{bmatrix} #1 \end{bmatrix}^T} \newcommand{\Reals}{\mathbf{R}} \newcommand{\RR}{\mathbf{R}} \newcommand{\Complex}{\mathbf{C}} \newcommand{\Field}{\mathbf{F}} \newcommand{\Pb}{\operatorname{Pr}} \newcommand{\E}[1]{\operatorname{E}[#1]} \newcommand{\Var}[1]{\operatorname{Var}[#1]} \newcommand{\argmin}[2]{\underset{#1}{\operatorname{argmin}} {#2}} \newcommand{\optmin}[3]{ \begin{align*} & \underset{#1}{\text{minimize}} & & #2 \\ & \text{subject to} & & #3 \end{align*} } \newcommand{\optmax}[3]{ \begin{align*} & \underset{#1}{\text{maximize}} & & #2 \\ & \text{subject to} & & #3 \end{align*} } \newcommand{\optfind}[2]{ \begin{align*} & {\text{find}} & & #1 \\ & \text{subject to} & & #2 \end{align*} } $$
A hyperplane is a set
where is nonzero and . The vector is referred to as the normal vector of the hyperplane, since, if is any point such that , the hyperplane can be expressed as
The Euclidean projection of a point onto the hyperplane is
One way to derive this solution is to analytically solve the optimization problem
using, for example, the method of Lagrange multiplers.
Consider two parallel hyperplanes and . The distance between the hyperplanes can be computed by projecting any point in the former hyperplane onto the latter hyperplane. In particular, the projection of onto is
Hence the distance between the two hyperplanes is
As an example, the width of the slab
is the distance between the hyperplanes and , which equals .