Theorem 13.24 — the closed-form convex hull of a union of two polymatroids

This is Theorem 13.24 of Balas's Disjunctive Programming, the goal theorem of this mission and the closing result of the entire book: a fully explicit, closed-form description of the convex hull of a union of two polymatroids, in the original variable space (no lifting, no auxiliary variables).

For polymatroid rank functions $r_1,r_2$ (satisfying $r(\emptyset)=0$, monotone, submodular),

The book's proof, resting on Propositions 13.22 and 13.23, reduces to examining the extreme points of $U$: a basic feasible solution has at most two nonzero components $u_A,u_B$. A single nonzero $u_A = 1/\max\{r_1(A),r_2(A)\}$ yields the first family of inequalities; two simultaneously nonzero $u_A,u_B$ solving $u_Ar_1(A)+u_Br_1(B)=1$, $u_Ar_2(A)+u_Br_2(B)=1$ have a (unique, positive) solution exactly when $(r_1(A)-r_2(A))(r_1(B)-r_2(B))<0$, yielding the second family. This generalizes an earlier result on the disjunction of matroid polyhedra (proved by different means) to arbitrary polymatroids.

Formalization Note. IsPolymatroidRankFunction (not IsApp1SetFunction) is the correct hypothesis here, matching the book's own "$r_i:2^N\to\mathbb R$ for $i=1,2$ are polymatroid rank functions (satisfying 1 and 3 in Application 1 and submodular)" — this is r(\emptyset)=0 plus monotonicity plus submodularity, not the r(A)\le|A| bound of IsApp1SetFunction (that bound is specific to the matroid-truncated special case of §13.4, not required for the general polymatroid closed form). Confusing the two hypotheses here would misstate exactly which set functions the theorem covers — this is the distinction BRIEF.md explicitly warns to get right.