Strong uniqueness, Lipschitz continuity, and continuous selections for metric projections in L1

AbstractThe relations between the lower semicontinuity of the metric projection PG onto a finite-dimensional subspace G of L1, the Lipschitz continuity of PG, the existence of continuous selections for PG, and uniform strong uniqueness of PG are studied. In particular, the lower semicontinuity of PG, the Lipschitz continuity of PG, and the uniform strong uniqueness of PG are all equivalent. If PG is lower semicontinuous, then PG has a Lipschitz continuous selection. Moreover, if G is one-dimensional, PG has a continuous selection if and only if it has a Lipschitz continuous selection