  We first consider a question raised by Alexander Eremenko and show that if
$\Omega $ is an arbitrary connected open cone in ${\mathbb R}^d$, then any two
positive harmonic functions in $\Omega $ that vanish on $\partial \Omega $ must
be proportional -an already known fact when $\Omega $ has a Lipschitz basis or
more generally a John basis. It is also shown however that when $d \geq 4$,
there can be more than one Martin point at infinity for the cone though
non-tangential convergence to the canonical Martin point at infinity always
holds. In contrast, when $d \leq 3$, the Martin point at infinity is unique for
every cone. These properties connected with the dimension are related to
well-known results of M. Cranston and T. R. McConnell about the lifetime of
conditioned Brownian motions in planar domains and also to subsequent results
by R. Ba\~nuelos and B. Davis. We also investigate the nature of the Martin
points arising at infinity as well as the effects on the Martin boundary
resulting from the existence of John cuts in the basis of the cone or from
other regularity assumptions. The main results together with their proofs
extend to cylinders ${\mathcal C}_Y(\Sigma)= {\mathbb R} \times \Sigma$ -where
$\Sigma $ is a relatively compact region of a manifold $M$-, equipped with a
suitable second order elliptic operator.
