The dimension-free threshold for maximal sections of ₚ

We prove the existence of a sharp threshold p₀ ≈ 26.265…, such that for every n and p≥ p₀, the volume maximizing section of the unit ball of ₚⁿ is the section with Ball's hyperplane (e₁+e₂√2)^⊥, whereas for p∈(2,p₀) the asymptotic maximizer as the ambient dimension n→∞ is the hyperplane orthogonal to the diagonal vector e₁+⋯+eₙ√n.