  We give new characterizations for the class of uniformly dense matroids and
study applications of these characterizations to graphic and real representable
matroids. We show that a matroid is uniformly dense if and only if its base
polytope contains a point with constant coordinates. As a main application, we
derive new spectral, structural and classification results for uniformly dense
graphs. In particular, we show that connected regular uniformly dense graphs
are $1$-tough and thus contain a (near-)perfect matching. As a second
application, we show that uniform density can be recognized in polynomial time
for real representable matroids, which includes graphs. Finally, we show that
strictly uniformly dense real represented matroids can be represented by
projection matrices with a constant diagonal and that they are parametrized by
a subvariety of the Grassmannian.
