  The spectral gap of the Neumann and Dirichlet Laplacians are each known to
have a sharp positive lower bound among convex domains of a given diameter.
Between these cases, for each positive value of the Robin parameter an
analogous sharp lower bound on the spectral gap is conjectured. In this paper
we show the extension of this conjecture to negative Robin parameters fails
completely, by proving the spectral gap of an explicit family of domains can be
exponentially small.
