  In this paper we study heights on quotient varieties in the sense of
Geometric Invariant Theory (GIT). We generalise a construction of Burnol and we
generalise diverse lower bounds of the height of semi-stable points due to
Bost, Zhang, Gasbarri and Chen. In order to prove Burnol's formula for the
height on the quotient we develop a Kempf-Ness theory in the setting of
Berkovich analytic spaces, completing the former work of Burnol.
