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Communications in Mathematical Physics

, Volume 319, Issue 1, pp 111–145 | Cite as

The Fundamental Gap of Simplices

  • Zhiqin Lu
  • Julie RowlettEmail author
Open Access
Article
  • 476 Downloads

Abstract

The gap function of a domain \({\Omega \subset \mathbb{R}^n}\) is
$$\xi(\Omega) := d^2 (\lambda_2 - \lambda_1)$$
, 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 \({\Omega \subset \mathbb{R}^n}\),
$$\xi(\Omega) \geq 3 \pi^2$$
, 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 \({T \subset \mathbb{R}^2}\),
$$\xi(T) \geq \frac{64 \pi^2}{9}$$
, with equality if and only if T is equilateral.

Keywords

Modulus Space Variational Principle Equilateral Triangle Convex Domain Cauchy Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaIrvineUSA
  2. 2.Max Planck Institut für MathematikBonnGermeny

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