Arbitrarily large neighborly families of symmetric convex polytopes

A family of convex d-polytopes in E _d is called neighborly if every two of them have a (d–1)-dimensional intersection. Settling an old problem of B. Grünbaum, we show that there exist arbitrarily large neighborly families of centrally (or any other prescribed type of) symmetric convex d-poliytopes in E _d ,for all d3; moreover, they can all be congruent, if d4.A version of this paper has been written while the author visited R. K. Guy in Calgary, Alberta, Canada, in the summer of 1981; the author wishes to thank Louise and Richard Guy for their warm hospitality.