Revisiting Tietze–Nakajima: Local and Global Convexity for Maps

Abstract

AbstractA theorem of Tietze and Nakajima, from 1928, asserts that if a subset X of ℝn is closed, connected, and locally convex, then it is convex. We give an analogous “local to global convexity” theorem when the inclusion map of X to ℝn is replaced by a map from a topological space X toℝRn that satisfies certain local properties. Our motivation comes from the Condevaux–Dazord–Molino proof of the Atiyah–Guillemin–Sternberg convexity theorem in symplectic geometry.