On Ikebe's criterion (Q582527)

A 0-2 law for the metric projection is shown to hold on most of the common Banach spaces. A normed space E has property \((Ik^ 1_ 1)\) provided that every nontrivial segment in any face of the unit ball of E has a parallel segment of length 2 in the same face. This property implies that any subspace V of E satisfying \[ \| v\| <2\| x\| \quad for\quad all\quad x\in E,\quad v\in P_ Vx \] (P\({}_ Vx=elements\) of best approximation to x in V) is semichebyshev (i.e. \(P_ Vx\) consists of at most one element). It is shown that \(L_ 1(\mu)\), C(K), \(C_ 0(Q)\) and more generally all Lindenstrauss spaces have property \((Ik^ 1_ 1)\). Some other examples combining these with strictly convex norms are studied.