Abstract: We give a counterexample to the long standing conjecture
that the ball maximises the first eigenvalue of
the Robin eigenvalue problem with negative boundary parameter
among domains of the same volume.
Furthermore, we show that the conjecture holds in two dimensions
provided that the boundary parameter is small.
This is the first known example
within the class of isoperimetric spectral problems
for the first eigenvalue of the Laplacian where the ball
is not an optimiser.