Let Ω = R n ∖ E \Omega = {\mathbb {R}^n}\backslash E , where E E is a closed subset of the hyperplane { x n = 0 } \left \{ {{x_n} = 0} \right \} and every point of E E is regular for the Dirichlet problem on Ω \Omega . Further, let α k {\alpha _k} . denote the ( n − 1 ) (n - 1) -dimensional measure of the set { X ∈ Ω : x n = 0 , e k > | X | > e k + 1 } \{ X \in \Omega :{x_n} = 0,{e^k} > |X| > {e^{k + 1}}\} . It is known that the cone, P E {\mathcal {P}_E} , of positive harmonic functions on Ω \Omega which vanish on E E has dimension 1 or 2. In this paper it is shown that if ∑ e − n k α k n / ( n − 1 ) > + ∞ \sum {{e^{ - nk}}\alpha _k^{n/(n - 1)} > + \infty } then dim P E = 2 \dim {\mathcal {P}_E} = 2 . This result, which in the case n = 2 n = 2 implies a recent theorem of Segawa, is also shown to be sharp.