  We consider $n$-dimensional spacetimes which are axisymmetric--but not
necessarily stationary (!)--in the sense of having isometry group $U(1)^{n-3}$,
and which satisfy the Einstein equations with a non-negative cosmological
constant. We show that any black hole horizon must have area $A \ge 8\pi |J_+
J_-|^\half$, where $J_\pm$ are distinguished components of the angular momentum
corresponding to linear combinations of the rotational Killing fields that
vanish somewhere on the horizon. In the case of $n=4$, where there is only one
angular momentum component $J_+=J_-$, we recover an inequality of 1012.2413
[gr-qc]. Our work can hence be viewed as a generalization of this result to
higher dimensions. In the case of $n=5$ with horizon of topology $S^1 \times
S^2$, the quantities $J_+=J_-$ are the same angular momentum component (in the
$S^2$ direction). In the case of $n=5$ with horizon topology $S^3$, the
quantities $J_+, J_-$ are the distinct components of the angular momentum. We
also show that, in all dimensions, the inequality is saturated if the metric is
a so-called ``near horizon geometry''. Our argument is entirely quasi-local,
and hence also applies e.g. to any stably outer marginally trapped surface.
