Neighborly Embedded Manifolds

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 )≤2 k ( k −1) n . Related results and open problems are discussed.