Almost convex metrics and Peano compactifications

Let ( X , d ) denote a locally connected, connected separable metric space. We say the X is S - metrizable provided there is a topologically equivalent metric ρ on X such that ( X , ρ ) has Property S , i.e., for any ϵ > 0 , X is the union of finitely many connected sets of ρ -diameter less than ϵ . It is well-known that S -metrizable spaces are locally connected and that if ρ is a Property S metric for X , then the usual metric completion ( X ˜ , ρ ˜ ) of ( X , ρ ) is a compact, locally connected, connected metric space; i.e., ( X ˜ , ρ ˜ ) is a Peano compactification of ( X , ρ ) . In an earlier paper, the author conjectured that if a space ( X , d ) has a Peano compactification, then it must be S -metrizable. In this paper, that conjecture is shown to be false; however, the connected spaces which have Peano compactificatons are shown to be exactly those having a totally bounded, almost convex metric. Several related results are given.