Abstract
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 if P 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 dim M < ∞ 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.
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