A theorem of Tietze and Nakajima, from 1928, asserts that if a subset X of R is closed, connected, and locally convex, then it is convex [Ti, N]. There are many generalizations of this “local to global convexity” phenomenon in the literature; a partial list is [BF, C, Ka, KW, Kl, SSV, S, Ta]. This paper contains an analogous “local to global convexity” theorem when the inclusion map of X to R i...