Does a matroid base polytope contain its circumcenter?

Question

Let $(X,\mathcal B)$ be a matroid on the ground set $X=(x_1,...,x_n)$ and with set of bases $\mathcal B$, and let $P\subset\Bbb R^n$ be its matroid base polytope (i.e. the convex hull of the characteristic vectors of the bases $B\in\mathcal B$). The circumcenter of $P$ is the unique point $p\in\mathrm{aff}(P)$ that has the same distance to all vertices of $P$.

Question: Does $P$ contain its circumcenter, perhaps even in its relative interior?

Here relative interior means the interior of $P$ considered as a subset of its affine hull.

Example. This is true for uniform matroids as their matrix base polytopes are hypersimplices.

Answer 1

I do not think so. Consider a uniform matroid on 3 elements $a, b, c$ of rank 2, and take 100 copies of $a$ (so, totally we have 102 elements). Then the matroid base polytope has full dimension 101, thus the circumcentre has all coordinates $1/51$. But each base has the sum of coordinates corresponding to $b$ and $c$ not less than 1, thus so does any their convex combination.