Abstract
We study the function KVol : (X, ω) → Vol(X, ω) sup α,β Int(α,β) lg (α)lg (β) defined on the moduli spaces of translation surfaces. More precisely, let Tn be the Teichmüller discs of the original Veech surface (Xn, ωn) arising from right-angled triangle with angles (π/2, π/n, (n − 2)π/2n) by the unfolding construction for n ≥ 5. For n ≡ 2 mod 4 and any (X, ω) ∈ Tn, we establish the (sharp) bounds a(n) 4 cot π n ≤ KVol(X, ω) ≤ a(n) 4 cot π n • 1 sin 2π a(n) , where a(n) = 2n if n is odd, and a(n) = n otherwise. The lower bound is uniquely realized at (Xn, ωn).