The Fundamental Gap of Simplices

The gap function of a domain Ω ⊂ Rn is ξ(Ω) := d(λ2 − λ1), where d is the diameter ofΩ , andλ1 andλ2 are the first two positive Dirichlet eigenvalues of the Euclidean Laplacian on Ω . It was recently shown by Andrews and Clutterbuck (J Amer Math Soc 24:899–916, 2011) that for any convex Ω ⊂ Rn , ξ(Ω) ≥ 3π2, where the infimum occurs for n = 1. On the other hand, the gap function on the moduli space of n-simplices behaves differently. Our first theorem is a compactness result for the gap function on the moduli space of n-simplices. Next, specializing to n = 2, our second main result proves the recent conjecture of Antunes-Freitas (J Phys A: Math Theor 41(5):055201, 2008) for any triangle T ⊂ R2, ξ(T ) ≥ 64π 2 9 , with equality if and only if T is equilateral.