We give new characterizations for the class of uniformly dense matroids, and we describe applications 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, and if and only if there exists a measure on the bases such that every element of the ground set has equal probability to be in a random basis with respect to this measure. As one application, we derive new spectral, structural and classification results for uniformly dense graphic matroids. 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 strictly uniformly dense real representable matroids can be represented by projection matrices with constant diagonal and that they are parametrized by a subvariety of the real Grassmannian.

翻译：我们给出了均匀密度拟阵类的新的刻画，并描述了这些结果在图拟阵和实可表示拟阵中的应用。我们证明：一个拟阵是均匀密度的当且仅当其基多面体包含一个坐标全为常数的点，也当且仅当在基上存在一个测度，使得在按此测度随机选取的基中，底集的每个元素具有相等的出现概率。作为第一个应用，我们得到了均匀密度图拟阵的新谱性质、结构性质和分类结果。特别地，我们证明了连通正则均匀密度图是1-坚韧的，因此包含一个（近）完美匹配。作为第二个应用，我们证明了严格均匀密度的实可表示拟阵可由具有常数对角元的投影矩阵表示，并且它们由实格拉斯曼流形的子簇参数化。