I'm interested in connected matroids $M$ on the ground set $[n]$ for which there is no connected matroid on $[n]$ of the same rank but with a strictly smaller set of bases (by inclusion). Equivalently, the matroid polytope of $M$ has dimension $n-1$ and does not contain any other matroid polytope of dimension $n-1$. Obvious examples are uniform matroids of ranks $1$ and $n-1$. A less trivial example for $n=4$ is a matroid of rank 2 with 5 bases.


What is known about such matroids? Is there some classification or interesting alternative characterization?



"Minimal" connected matroids
In general there does not seem to be a nice classification. You're asking for connected matroids of a given rank which are minimal in the "weak map order" (a weak map between matroids of the same rank is an inclusion of bases). In [Weak maps of combinatorial geometries, Theorem 6.10](https://www.ams.org/journals/tran/1975-206-00/S0002-9947-1975-0371693-2/S0002-9947-1975-0371693-2.pdf), Dean Lucas shows that every connected binary matroid is minimal. In particular, every connected graphic matroid is minimal. I believe there are examples which are not binary.


