Neighborly polytope

In geometry and polyhedral combinatorics, a k-neighborly polytope is a convex polytope in which every set of k or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which every pair of vertices is connected by an edge, forming a complete graph. 2-neighborly polytopes with more than four vertices may exist only in spaces of four or more dimensions, and in general a k-neighborly polytope (other than a simplex) requires a dimension of 2k or more. A d-simplex is d-neighborly. A polytope is said to be neighborly, without specifying k, if it is k-neighborly for ${\displaystyle k=\lfloor d/2\rfloor }$. If we exclude simplices, this is the maximum possible k: in fact, every polytope that is k-neighborly for some ${\displaystyle k\geq 1+\lfloor d/2\rfloor }$ is a simplex.[1]

In a k-neighborly polytope with k ≥ 3, every 2-face must be a triangle, and in a k-neighborly polytope with k ≥ 4, every 3-face must be a tetrahedron. More generally, in any k-neighborly polytope, all faces of dimension less than k are simplices.

The cyclic polytopes formed as the convex hulls of finite sets of points on the moment curve (t, t², ..., t^d) in d-dimensional space are automatically neighborly. Theodore Motzkin conjectured that all neighborly polytopes are combinatorially equivalent to cyclic polytopes.[2] However, contrary to this conjecture, there are many neighborly polytopes that are not cyclic: the number of combinatorially distinct neighborly polytopes grows superexponentially, both in the number of vertices of the polytope and in the dimension.[3]

The convex hull of a set of random points, drawn from a Gaussian distribution with the number of points proportional to the dimension, is with high probability k-neighborly for a value k that is also proportional to the dimension.[4]

The number of faces of all dimensions of a neighborly polytope in an even number of dimensions is determined solely from its dimension and its number of vertices by the Dehn–Sommerville equations: the number of k-dimensional faces, f_k, satisfies the inequality

${\displaystyle f_{k-1}\leq \sum _{i=0}^{d/2}{}^{*}\left({\binom {d-i}{k-i}}+{\binom {i}{k-d+i}}\right){\binom {n-d-1+i}{i}},}$

where the asterisk means that the sums ends at ${\displaystyle i=\lfloor d/2\rfloor }$ and final term of the sum should be halved if d is even.[5] According to the upper bound theorem of McMullen (1970),[6] neighborly polytopes achieve the maximum possible number of faces of any n-vertex d-dimensional convex polytope.

A generalized version of the happy ending problem applies to higher-dimensional point sets, and implies that for every dimension d and every n > d there exists a number m(d,n) with the property that every m points in general position in d-dimensional space contain a subset of n points that form the vertices of a neighborly polytope.[7]

References

1. Grünbaum, Branko (2003), Kaibel, Volker; Klee, Victor; Ziegler, Günter M. (eds.), Convex Polytopes, Graduate Texts in Mathematics, 221 (2nd ed.), Springer-Verlag, p. 123, ISBN 0-387-00424-6.
2. Gale, David (1963), "Neighborly and cyclic polytopes", in Klee, Victor (ed.), Convexity, Seattle, 1961, Symposia in Pure Mathematics, 7, American Mathematical Society, pp. 225–233, ISBN 978-0-8218-1407-9.
3. Shemer, Ido (1982), "Neighborly polytopes" (PDF), Israel Journal of Mathematics, 43 (4): 291–314, doi:10.1007/BF02761235.
4. Donoho, David L.; Tanner, Jared (2005), "Neighborliness of randomly projected simplices in high dimensions", Proceedings of the National Academy of Sciences of the United States of America, 102 (27): 9452–9457, doi:10.1073/pnas.0502258102, PMC 1172250, PMID 15972808.
5. Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, pp. 254–258, ISBN 0-387-94365-X.
6. McMullen, Peter (1970), "The maximum numbers of faces of a convex polytope", Mathematika, 17: 179–184, doi:10.1112/S0025579300002850.
7. Grünbaum, Branko (2003), Kaibel, Volker; Klee, Victor; Ziegler, Günter M. (eds.), Convex Polytopes, Graduate Texts in Mathematics, 221 (2nd ed.), Springer-Verlag, p. 126, ISBN 0-387-00424-6. Grünbaum attributes the key lemma in this result, that every set of d + 3 points contains the vertices of a (d + 2)-vertex cyclic polytope, to Micha Perles.