On the Spectral Gap for Convex Domains
  We prove the following for a bounded convex planar domain that is symmetric with respect to both coordinate axes. Consider a centered rectangle with sides parallel to the axes that strictly contains the domain. If the domain is not a certain kind of rectangle, the spectral gap of the domain is larger than the spectral gap of the rectangle. We also provide explicit lower bounds for the differnce between the gaps. 
Burgess Davis and Majid Hosseini