Free products, extremal matroids, and a generalization of perfect matroid designs

Extremal matroids are those that yield vertices in the polytope of matroids. We show how to get the point, in the appropriate polytope of matroids, for the free product $M\square M'$ of matroids $M$ and $M'$ from the points for $M$ and $M'$ in their polytopes of matroids. With this we show that if $M\square M'$ is extremal, then $M$ and $M'$ are extremal. The converse is false. We identify a large class of matroids that includes perfect matroid designs and sparse paving matroids, and we find sufficient conditions under which free products of extremal matroids in this class are extremal.