Characterizations of continuous and Lipschitz continuous metric selections in normed linear spaces

Characterizations are given of when the metric projection P_M onto a proximal subspace M has a continuous, pointwise Lipschitz continuous, or Lipschitz continuous selection. Moreover, it is shown that ifP_M has a continuous selection, then it has one which is also homogeneous and additive modulo M. An analogous result holds if P_M has a pointwise Lipschitz or Lipschitz continuous selection provided that M is complemented. If dimM < ∞ and P_M is Lipschitz (resp. pointwise Lipschitz) continuous, then P_M has a Lipschitz (resp. pointwise Lipschitz) continuous selection. A conjecture of R. Holmes and B. Kripke (Michigan Math. J. 15 (1968), 225–248) is resolved.