  We compute the hessian of the natural Hermitian form successively on the
Calabi family of a hyperk\"ahler manifold, on the twistor space of a
4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a
quaternionic K\"ahler manifold. We show a strong convexity property of the
cycle space of twistor lines on the Calabi family of a hyperk\"ahler manifold.
We also prove convexity properties of the 1-cycle space of the twistor space of
a 4-dimensional anti-self-dual Einstein manifold of non-positive scalar
curvature and of the 1-cycle space of the twistor space of a quaternionic
K\"ahler manifold of non-positive scalar curvature. We check that no
non-K\"ahler strong KT manifold occurs as such a twistor space.
