Abstract An embedding of an n-dimensional manifold M into R d is called k-neighborly if, for every k points on the embedded manifold, there is a hyperplane H in R d which supports the manifold precisely at these points. Micha A. Perles (Problems presented in Oberwolfach conference on “Convexity”, [1982]) asked: What is the smallest dimension d(k,n) of the ambient space in which a k-neighborly n-dimensional manifold exists? We prove that d(k,n)≤2k(k−1)n. Related results and open problems are discussed.