Let P be a polytope defined by the system A x b, where A R^m n, b Rᵐ, and rank (A) = n. We give a short geometric proof of the following tight upper bound on the number of vertices of P: n! ΔΔ₀ₕ₄ₑ₀₆₄ vol (B₂) 1πn (2 πe) ^n/2 n^n/2 ΔΔ₀ₕ₄ₑ₀₆₄, where Δ is the maximum absolute value of n n subdeterminants of A, and Δ₀ₕ₄ₑ₀₆₄ is the average absolute value of subdeterminants of A corresponding to a triangulation of P's normal fan. Assuming that A is integer, such polyhedra are called Δ-modular polyhedra. Note that in the integer case, the bound can be simplified via the inequality Δ₀ₕ₄ₑ₀₆₄ Δ_ 1, where Δ_ is the minimum absolute value of subdeterminants of A corresponding to feasible bases of A x b. For this, we prove and use a symmetric variant of Macbeath's theorem. Additionally, we give a direct argument based on prior results in the field, showing that the graph diameter of P is bounded by O (n³ ΔΔ_{} (n ΔΔ_{}) ). Thus, both characteristic of P are linear in Δ/Δ_. From an algorithmic perspective, we demonstrate that: Given A Q^m n, b Qᵐ, and an initial feasible solution to A x b, the convex hull of P can be constructed in O (n) ^n/2 m² ΔΔ₀ₕ₄ₑ₀₆₄ operations. For simple polyhedra, the dependence on m reduces to linear; Given A Z^m n and b Qᵐ, the number |P Zⁿ| can be computed in O (n) ⁿ Δ⁴Δ₀ₕ₄ₑ₀₆₄ arithmetic operations.
Mikhail et al. (Sun,) studied this question.