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The fundamental gap of simplices

Journal article, 2013

The gap function of a domain Ω⊂Rn is
ξ(Ω):=d2(λ2−λ1)
, where d is the diameter of Ω, and λ1 and λ2 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)≥64π29
, with equality if and only if T is equilateral.