Theorem 5.1 at the argmax set — the localisation of ω° at p₀ is matroidal iff argmax(ω[−p₀]) is an integral base set

Murota 1996, p. 291, Theorem 5.1, applied at the argmax set of the localisation's subgradient functional. Let $B \subseteq \mathbb{Z}^V$ be a finite integral base set, $\omega : \mathbb{Z}^V \to \mathbb{R}$, and $p_0 \in \mathbb{R}^V$. Write $\omega[-p_0](x) = \omega(x) - \langle p_0, x\rangle$ and $A_{p_0} = \operatorname{argmax}(\omega[-p_0]) = \{x \in B \mid \omega(x) - \langle p_0, x\rangle \ge \omega(y) - \langle p_0, y\rangle\ \forall y \in B\}$. Then the localisation $\hat{L}(\omega^\circ, p_0)(p) = \inf\{\langle p, b\rangle \mid b \in \partial\omega^\circ(p_0)\}$ of the concave conjugate $\omega^\circ$ at $p_0$ is \emph{matroidal} -- positive homogeneous and satisfying (C1) supermodularity and (C2) greediness -- if and only if $A_{p_0}$ is a finite integral base set.\n\nThe two ingredients are visible in Murota's proof. First, Eq. (5.12) identifies the localisation pointwise with the support function of $A_{p_0}$, $\min\{\langle p, x\rangle \mid x \in A_{p_0}\}$. Second, Lemma 4.3 says that every integer point of the convex hull of a maximiser set $A_{p_0}$ lies again in $A_{p_0}$, which is exactly the hypothesis that Theorem 5.1 requires of the set at which its support function is taken. Theorem 5.1 then turns (B1) for $A_{p_0}$ into matroidalness of that support function, hence of the localisation. This is the step that transports Theorem 5.1 to the conjugate and is the bridge between Eq. (5.12) and Theorem 4.4 in the proof of Theorem 5.3.

Retired. Retired because it quantified over every ω without the standing hypothesis of §4–§5: the argmax set can have holes, its support function (the localisation) can be matroidal while the set violates (B1) (accepted disproof); replaced by SteinitzExchange.LocalSupermod.matroidal_iff_argmax_v2, which assumes the argmax set hole-free as Theorem 5.1 requires.