Observations on the metric projection in finite dimensional Banach spaces

We consider the method of alternating (metric) projections for pairs of linear subspaces of finite dimensional Banach spaces. We investigate the size of the set of points for which this method converges to the metric projection onto the intersection of these subspaces. In addition we give a characterisation of the pairs of subspaces for which the alternating projection method converges to the projection onto the intersection for every initial point. We provide a characterisation of the linear subspaces of $\ell_p^n$, $1<p<\infty$, $p\neq 2$, which admit a linear metric projection and use this characterisation to show that in $\ell_p^3$, $1<p<\infty$, $p\neq 2$, the set of pairs of subspaces for which the alternating projection method converges to the projection onto the intersection is small in a probabilistic sense.