Black Holes Have No Short Hair

We show that in all theories in which black hole hair has been discovered the region with nontrivial structure of the nonlinear matter fields must extend beyond $3/2$ the horizon radius, independently of all other parameters present in the theory. We argue that this is a universal lower bound that applies in every theory where hair is present. This no short hair conjecture is then put forward as a more modest alternative to the original no hair conjecture, the validity of which now seems doubtful.