Polyhedral combinatorics of multi-index axial transportation problems

Infinitesimal semi‐invariant pictures and co‐amalgamation

“…\end{equation*}$$By Lemma 5.3.4, we need only show that there exist coefficients $$\lambda ^{\prime }_{{\mathcal {X}}}\geqslant 0$$ for each $$p({\mathcal {X}}) \in {\mathcal {P}}^+$$ so that $$\lambda _{i,j} = \lambda ^{\prime }_{i,j}$$ for all $$X_{j,1}^i \in \operatorname{tp}({\mathcal {P}}^+)$$. In general, the set of possible $$\lbrace \lambda ^{\prime }_{{\mathcal {X}}}\rbrace _{p({\mathcal {X}}) \in {\mathcal {P}}^+}$$ satisfying this property is an $$m$$ ‐index transportation polytope (see, e.g., [39, section 1]). As for $$i$$ fixed we have $$\sum _{j} \lambda _{i,j} = \sum _{p({\mathcal {X}}) \in {\mathcal {P}}^+}\lambda _{{\mathcal {X}}} \geqslant 0$$, which does not depend on $$i$$, this solution space is nonempty.…”