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**](https://en.wikipedia.org/wiki/Matroid_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 $P$.



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



Does a matroid base polytope contain its circumcenter?
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.


