Neighborly polytopes

Abstract

A 2m-polytopeQ isneighborly if eachm vertices ofQ determine a face. It is shown that the combinatorial structure of a neighborly 2m-polytope determines the combinatorial structure of every subpolytope. We develop a construction of “sewing a vertex onto a polytope”, which, when applied to a neighborly 2m-polytope, yields a neighborly 2m-polytope with one more, vertex. Using this construction, we show that the numberg(2m+β,2m) of combinatorial types of neighborly 2m-polytopes with 2m+β vertices grows superexponentially as β→∞ (m≧2 fixed) and asm→∞ (β≧4 fixed).