We first consider the additive Brownian motion process (X(s(1), s(2)), (s(1), s(2)) is an element of R-2) defined by X(s(1), s(2)) = Z(1)(s(1)) - Z2(s2), where Z(1) and Z(2) are two independent (two-sided) Brownian motions. We show that with probability 1, the Hausdorff dimension of the boundary of any connected component of the random set {(s(1,) s(2)) is an element of R-2 : X(s(1), s(2)) > 0} is equal to

1/4 (1 + root 13 + 4 root 5) similar or equal to 1.421.

Then the same result is shown to hold when X is replaced by a standard Brownian sheet indexed by the non-negative quadrant.