Talk:Matroid polytope

Incorrect set-builder description of the Matroid polytope

$P_{M}:=\left\{{\textbf {x}}\in \mathbb {R} _{+}^{E}~|~\sum _{e\in U}{\textbf {x}}(e)\leq r(U),\forall U\subseteq E\right\}$

is an incorrect description of the matroid basis polytope. Rather, it describes the matroid independence polytope. This can be easily seen as ${\textbf {x}}={\textbf {0}}$, the characteristic vector of the empty set, satisfies $0\leq r(U),~\forall U\subseteq E$. The correct description would be

$P_{M}:=\left\{{\textbf {x}}\in \mathbb {R} _{+}^{E}~|~\sum _{e\in U}{\textbf {x}}(e)\leq r(U),~\forall U\subseteq E,~\sum _{e\in E}{\textbf {x}}(e)=r\right\}$.

NaturalLogZ (talk) 02:56, 3 October 2024 (UTC)