A theorem of Tietze and Nakajima, from 1928, asserts that if a subset $X$ of $\mathbb{R}^n$ 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 $\mathbb{R}^n$ is replaced by a map from a topological space $X$ to $\mathbb{R}^n$ that satisfies certain local properties. Our motivation comes from the Condevaux--Dazord--Molino proof of the Atiyah--Guillemin--Sternberg convexity theorem in symplectic geometry.

This paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a \emph{polytope} if each of its finite-dimensional affine sections is a (standard) polytope. We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open).