A refined lower bound theorem for d $d$ d-polytopes with at most 2 d $2d$ 2 d vertices

Source abstract

Abstract In 1967, Grünbaum conjectured that the function ϕ k ( d + s , d ) := ( d + 1 k + 1 ) + ( d k + 1 ) − ( d + 1 − s k + 1 ) , for 2 ⩽ s ⩽ d $$\begin{align*}\phi_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for } 2\leqslant s\leqslant d \end{align*}$$ provides the minimum number of k $k$ k -faces for a d $d$ d -dimensional polytope (abbreviated as a d $d$ d -polytope) with d + s $d+s$ d plus s vertices. In 2021, Xue proved this conjecture for each k ∈ [ 1 , d − 2 ] $k\in [1, d-2]$ k element of left bracket 1 comma d minus 2 right bracket and characterized the unique minimizers, each having d + 2 $d+2$ d plus 2 facets. In this article, we refine Xue’s theorem by considering d $d$ d -polytopes with d + s $d+s$ d plus s vertices ( 2 ⩽ s ⩽ d $2\leqslant s\leqslant d$ 2 less than or slanted equals s less than or slanted equals d ) and at least d + 3 $d+3$ d plus 3 facets. If s = 2 $s=2$ s equals 2 , then there is precisely one minimizer for many values of k $k$ k . For other values of s $s$ s , the number of k $k$ k -faces is at least ϕ k ( d + s , d ) + ( d − 1 k ) − ( d + 1 − s k ) $\phi _k(d+s,d)+\binom {d-1}{k}-\binom {d+1-s}{k}$ phi Subscript k Baseline left parenthesis d plus s comma d right parenthesis plus StartBinomialOrMatrix d minus 1 Choose k EndBinomialOrMatrix minus StartBinomialOrMatrix d plus 1 minus s Choose k EndBinomialOrMatrix , which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of s $s$ s and k $k$ k . We also characterize the minimizing polytopes.