Laplacian and spectral gap in regular Hilbert geometries
  We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues. 
Thomas Barthelm\'e, Bruno Colbois, Micka\"el Crampon and Patrick   Verovic
math.DG math.SP