# Opposite vertices of base polyhedra

Is it true that if all vertices of a base polyhedron *B* are in [math]\{0,1,-1\}^n[/math] and [math]0 \in B[/math], then *B* has a vertex *v* such that *-v* is also a vertex?

## Remarks

This would imply the following conjecture of Frank:

**Conjecture.** If *P* is an integer 2-polymatroid which contains the **1** (all-ones) vector, then it has a vertex *v* such that **2**-*v* is in *P*.

It is known by the Matroid union theorem that if the all-1/2 vector is in a matroid polyhedron, then the ground set of the matroid can be covered by 2 bases. This implies that the above conjecture is true if every vertex of *P* has only even coordinates.

A possible strengthening is the following.

**Question.** Is it true that if all vertices of an integer g-polymatroid *Q* are in [math]\{0,1,-1\}^n[/math] and [math]0 \in Q[/math], then *Q* has a vertex *v* such that *-v* is in *Q*?