---
tag: 9V4PR1RVXW
kind: concept
name: Neighborly polytope
msc: [52B05, 52B11, 52B12]
specializes: [300K6CB1KH]
sources:
  - url: https://en.wikipedia.org/wiki/Neighborly_polytope
same_as:
  wikidata: Q17099348
  wikipedia: Neighborly polytope
---

## Summary

A [[convex polytope|300K6CB1KH]] is $k$-neighborly when every set of $k$ of its [[vertices|R7V3S9V891]] spans a face.
A $d$-polytope is called neighborly when it is $\floor{d/2}$-neighborly, the largest value possible for a polytope that is not a simplex.
Neighborly polytopes exist in every dimension and with any number of vertices beyond $d$.
Neighborly simplicial polytopes have the largest possible numbers of faces, which is why they appear in the upper bound theorem.

## Explanations

### L1 human

In a polygon or a solid in space, two corners are usually not joined by an edge.
In four or more dimensions this can change: there are four-dimensional solids where every pair of corners is joined by an edge.
Such shapes are called neighborly, because every corner is a neighbour of every other.
In higher dimensions the same happens with triples, quadruples and so on of corners.