In 1967, Grünbaum conjectured that the functionprovides the minimum number of -faces for a -dimensional polytope (abbreviated as a -polytope) with + vertices.In 2021, Xue proved this conjecture for each ∈ 1, -2 and characterised the unique minimisers, each having + 2 facets.In this paper, we refine Xue's theorem by considering -polytopes with + vertices (2 ⩽ ⩽ ) and at least + 3 facets.If = 2, then there is precisely one minimiser for many values of .For other values of , the number of -faces is at least ( + , ) + -1 -+1- , which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of and .We also characterise the minimising polytopes.
Wang et al. (Mon,) studied this question.