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

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.