A sharp bound on the size of a connected matroid

This paper proves that a connected matroid M in which a largest circuit and a largest cocircuit have c and c* elements, respectively, has at most 2 cc* elements. It is also shown that if e is an element of M and ce and c* are the sizes of a largest circuit containing e and a largest cocircuit containing e, then IE(M)I < (Ce-1)(Ce -1)+1. Both these bounds are sharp and the first is proved using the second. The second inequality is an interesting companion to Lehman's width-length inequality which asserts that the former inequality can be reversed for regular matroids when ce and c* are replaced by the sizes of a smallest circuit containing e and a smallest cocircuit containing e. Moreover, it follows from the second inequality that if u and v are distinct vertices in a 2-connected loopless graph G, then IE(G)l cannot exceed the product of the length of a longest (u, v)-path and the size of a largest minimal edge-cut separating u from v.