Separated sequences in nonreflexive Banach spaces

We prove that there is c > 1 such that the unit ball of any nonreflexive Banach space contains a c-separated sequence. The supremum of these constants c is estimated from below by 5 root 4 and from above approximately by 1.71. Given any p >1, we also construct a nonreflexive space so that if the convex hull of a sequence is sufficiently close to the unit sphere, then its separation constant does not exceed 2(1/p).