English

A Proof of Gr\"unbaum's Lower Bound Conjecture for general polytopes

Combinatorics 2020-04-21 v1

Abstract

In 1967, Gr\"unbaum conjectured that any dd-dimensional polytope with d+s2dd+s\leq 2d vertices has at least ϕk(d+s,d)=(d+1k+1)+(dk+1)(d+1sk+1)\phi_k(d+s,d) = {d+1 \choose k+1 }+{d \choose k+1 }-{d+1-s \choose k+1 } kk-faces. We prove this conjecture and also characterize the cases in which equality holds.

Keywords

Cite

@article{arxiv.2004.08429,
  title  = {A Proof of Gr\"unbaum's Lower Bound Conjecture for general polytopes},
  author = {Lei Xue},
  journal= {arXiv preprint arXiv:2004.08429},
  year   = {2020}
}