Interpolate isoperimetric lower bounds for polytopes with m vertices where n+1 < m < 2n

Determine the optimal asymptotic lower bound for the isoperimetric quotient iq(K) of an n-dimensional convex polytope K with exactly m vertices, in the regime n+1 < m < 2n, thereby interpolating between the bounds iq(Δ_n) ≍ n for the regular simplex (m = n+1) and iq(B_{ℓ1}^n) ≍ √n for the cross-polytope (m = 2n).

Background

The classical isoperimetric inequality gives iq(K) ≳ √n for all convex bodies in ℝ^n. For polytopes, more structure can improve this bound depending on combinatorial constraints.

When K is a simplex (m = n+1 vertices), one has iq(K) ≍ n (e.g., for the regular simplex Δn). On the other hand, when m ≥ 2n, the n-dimensional cross-polytope B{ℓ1}^n has m = 2n vertices and satisfies iq(B_{ℓ1}^n) ≍ √n, showing that the general √n bound cannot be improved in that regime.

The paper highlights that the sharp behavior of the minimal possible isoperimetric quotient as a function of the number of vertices m is unknown in the intermediate range n+1 < m < 2n.