Polyhedral selection event and contrast interval

Fix integers $n\ge 1$ and $m\ge 0$, a matrix $A\in {\mathbb{R}}^{m\times n}$, a vector $b\in {\mathbb{R}}^m$, and a fixed nonzero vector $\eta \in {\mathbb{R}}^n$. Inequalities between vectors are coordinatewise. Define the projection matrix and contrast direction $P_{\eta }=\frac{\eta {\eta }^T}{\mathrm{\parallel }\eta {\mathrm{\parallel }}^2},c_{\eta }=\frac{\eta }{\mathrm{\parallel }\eta {\mathrm{\parallel }}^2}\mathrm{.}$ The matrix $P_{\eta }$ is the orthogonal projection onto $\mathrm{span}\{\eta \}$ (The orthogonal projection $P_{W}v$ is the $W$-component in $V=W\oplus W^{\perp }$). For a vector $y\in {\mathbb{R}}^n$, put $T(y)={\eta }^{T}y,Z(y)=(I_n-P_{\eta })y\mathrm{.}$ Then $y=c_{\eta }T(y)+Z(y)$ and ${\eta }^{T}Z(y)=0$.

For row $j$ of $A$, written $a_j^T$, define $r_j=a_j^T{c}_{\eta },d_j(z)=b_j-a_j^{T}z\mathrm{.}$ Let $J_{-}=\{j:r_j<0\}$, $J_{+}=\{j:r_j>0\}$, and $J_0=\{j:r_j=0\}$. Define $L(z)=\{\begin{array}{ll}{\max }_{j\in J_{-}}{d}_j(z)\mathrm{/}{r}_j,& J_{-}\mathrm{\ne }\mathrm{\varnothing },\\ -\mathrm{\infty },& J_{-}=\mathrm{\varnothing },\end{array}U(z)=\{\begin{array}{ll}{\min }_{j\in J_{+}}{d}_j(z)\mathrm{/}{r}_j,& J_{+}\mathrm{\ne }\mathrm{\varnothing },\\ +\mathrm{\infty },& J_{+}=\mathrm{\varnothing }\mathrm{.}\end{array}$ Define $G(z)$ to hold exactly when $d_j(z)\ge 0$ for every $j\in J_0$. The selection event is $S=\{y:Ay\le b\}$. The interval associated with $z$ is $I(z)=\mathrm{\varnothing }$ if $G(z)$ fails or $L(z)>U(z)$, and is $[L(z),U(z)]$ otherwise. The next lemma proves that this interval describes precisely the fiber of $S$ along the contrast direction.

The empty-row case $m=0$ gives $L=-\mathrm{\infty }$, $U=+\mathrm{\infty }$, and $G$ true. Infinite endpoints mean that the corresponding side is not truncated.