  We show that in all theories in which black hole hair has been discovered,
the region with non-trivial structure of the non-linear 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 {\it no short hair
conjecture} is then put forward as a more modest alternative to the now
debunked {\it no hair conjecture}.
