Error bound for a finite simplex and affine slice

For each support set $J\subseteq \{1,\dots ,d\}$ realized as $J=\mathrm{supp}q$ by some $q\in F$, define $C_J=\{v:{\sum }_i{v}_i=0,v_i\ge 0(i\mathrm{\notin }J),⟨v,w⟩\le 0\text{ for every }w\text{ with }{\sum }_i{w}_i=0,Mw=0,w_i\ge 0(i\mathrm{\notin }J)\}\mathrm{.}$ This is a closed cone, being an intersection of closed linear constraints and closed half-spaces. If $v\in C_J$ and $Mv=0$, then $v$ is an allowed $w$ in its defining inequality, hence $\mathrm{\parallel }v{\mathrm{\parallel }}_2^2\le 0$ and $v=0$.

The set $F={\mathrm{\Delta }}_d\cap M^{-1}\{b\}$ is nonempty, closed, and convex: in coordinates it is given by finitely many closed linear inequalities and affine equations. Thus [F1] applies. Suppose $p\mathrm{\notin }F$ and let $q$ be its nearest point in $F$. Put $J=\mathrm{supp}q$ and $v=(p-q)\mathrm{/}\mathrm{\parallel }p-q{\mathrm{\parallel }}_2$. Then ${\sum }_i{v}_i=0$ and $v_i\ge 0$ outside $J$, because $p_i\ge 0=q_i$ there.

For each realized $J$, the set $C_J\cap \{v:\mathrm{\parallel }v{\mathrm{\parallel }}_2=1\}$ is compact by [F3]. If nonempty, the continuous function $v\mapsto \mathrm{\parallel }Mv{\mathrm{\parallel }}_2$ has a positive minimum $c_J$: a zero would contradict step 1.1. There are finitely many $J$, so the minimum $c$ of these positive $c_J$ is positive whenever any such unit section is nonempty. If all are empty, suppose $p\mathrm{\notin }F$ and take its nearest point $q\in F$ by [F1]. With $J=\mathrm{supp}q$ and $v=(p-q)\mathrm{/}\mathrm{\parallel }p-q{\mathrm{\parallel }}_2$, every vector $w$ in the tangent conditions defining $C_J$ has $q+tw\in F$ for small $t>0$. [F2] then gives $⟨v,w⟩\le 0$, so $v\in C_J$ and $\mathrm{\parallel }v{\mathrm{\parallel }}_2=1$, a contradiction. Hence $F={\mathrm{\Delta }}_d$ and the claim is immediate.

If $w$ satisfies the linear and sign conditions in the definition of $C_J$, then $q+tw\in F$ for all sufficiently small $t>0$: the affine equations are preserved, coordinates outside $J$ remain nonnegative, and the finitely many positive coordinates of $q$ on $J$ remain positive. By [F2], $⟨p-q,tw⟩\le 0$, so $⟨v,w⟩\le 0$ and $v\in C_J$.

For the nontrivial case of step 2.1, homogeneity and its constant $c$ give $\mathrm{\parallel }Mp-b{\mathrm{\parallel }}_2=\mathrm{\parallel }M(p-q){\mathrm{\parallel }}_2=\mathrm{\parallel }p-q{\mathrm{\parallel }}_2\mathrm{\parallel }Mv{\mathrm{\parallel }}_2\ge c{\mathrm{dist}}_2(p,F)$. Taking $H=1\mathrm{/}c$ proves the result for $p\mathrm{\notin }F$; for $p\in F$ the left side is zero. In the empty-unit-section case from step 2.1, $F={\mathrm{\Delta }}_d$ and any $H>0$ works.