  For the representation of eigenstates on a Poincar\'e section at the boundary
of a billiard different variants have been proposed. We compare these
Poincar\'e Husimi functions, discuss their properties and based on this select
one particularly suited definition. For the mean behaviour of these Poincar\'e
Husimi functions an asymptotic expression is derived, including a uniform
approximation. We establish the relation between the Poincar\'e Husimi
functions and the Husimi function in phase space from which a direct physical
interpretation follows. Using this, a quantum ergodicity theorem for the
Poincar\'e Husimi functions in the case of ergodic systems is shown.
