  Various higher-dimensional black holes have been shown to be unstable by
studying linearized gravitational perturbations. A simpler method for
demonstrating instability is to find initial data that describes a small
perturbation of the black hole and violates a Penrose inequality. An easy way
to construct initial data is by conformal rescaling of the unperturbed black
hole initial data. For a compactified black string, we construct initial data
which violates the inequality almost exactly where the Gregory-Laflamme
instability appears. We then use the method to confirm the existence of the
"ultraspinning" instability of Myers-Perry black holes. Finally we study black
rings. We show that "fat" black rings are unstable. We find no evidence of any
rotationally symmetric instability of "thin" black rings.
