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A refined lower bound theorem for $d$
-polytopes with at most $2d$
vertices
- Part of
-
- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 01 June 2026, pp. 1-25
-
- Article
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In 1967, Grünbaum conjectured that the function
$$ \begin{align*}\phi_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for } 2\leqslant s\leqslant d \end{align*} $$
provides the minimum number of $k$
-faces for a $d$
-dimensional polytope (abbreviated as a $d$
-polytope) with $d+s$
vertices. In 2021, Xue proved this conjecture for each $k\in [1, d-2]$
and characterized the unique minimizers, each having $d+2$
facets. In this article, we refine Xue’s theorem by considering $d$
-polytopes with $d+s$
vertices ($2\leqslant s\leqslant d$
) and at least $d+3$
facets. If $s=2$
, then there is precisely one minimizer for many values of $k$
. For other values of $s$
, the number of $k$
-faces is at least $\phi _k(d+s,d)+\binom {d-1}{k}-\binom {d+1-s}{k}$
, which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of $s$
and $k$
. We also characterize the minimizing polytopes.
