Neighborly Cubical Polytopes

نویسندگان

  • Michael Joswig
  • Günter M. Ziegler
چکیده

Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical convex d-polytope C d whose r-skeleton is combinatorially equivalent to that of the n-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂C d of a neighborly cubical polytope C n d maximizes the f -vector among all cubical (d− 1)-spheres with 2 vertices. While we show that this is true for polytopal spheres if n ≤ d+1, we also give a counter-example for d = 4 and n = 6. Further, the existence of neighborly cubical polytopes shows that the graph of the n-dimensional cube, where n ≥ 5, is “dimensionally ambiguous” in the sense of Grünbaum. We also show that the graph of the 5-cube is “strongly 4-ambiguous”. In the special case d = 4, neighborly cubical polytopes have f3 = f0 4 log2 f0 4 vertices, so the facet-vertex ratio f3/f0 is not bounded; this solves a problem of Kalai, Perles and Stanley studied by Jockusch.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Neighborly Cubical Polytopes and Spheres

We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan [4]. By relating the two constructions we obtain an explicit description of a non-polytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of M...

متن کامل

Construction and Analysis of Projected Deformed Products

We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are “strictly preserved” under projection. Thus, starting from an arbitrary neighborly simplicial (d−2)-polytope...

متن کامل

Prodsimplicial-Neighborly Polytopes

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal & Ziegler’s “projecting deformed products” construction to products of arbitrary simple polytopes....

متن کامل

Many neighborly inscribed polytopes and Delaunay triangulations

We present a very simple explicit technique to generate a large family of point configurations with neighborly Delaunay triangulations. This proves that there are superexponentially many combinatorially distinct neighborly d-polytopes with n vertices that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, a...

متن کامل

Constructing neighborly polytopes and oriented matroids

A d-polytope P is neighborly if every subset of b d 2 c vertices is a face of P . In 1982, Shemer introduced a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept of neighborliness extends naturally to oriented matroids. Duals o...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Discrete & Computational Geometry

دوره 24  شماره 

صفحات  -

تاریخ انتشار 2000