  We make extensive numerical studies of masses and radii of proto-neutron
stars during the first second after their birth in core-collapse supernova
events. We use a quasi-static approach for the computation of proto-neutron
star structure, built on parameterized entropy and electron fraction profiles,
that are then evolved with neutrino cooling processes. We vary the equation of
state of nuclear matter, the proto-neutron star mass and the parameters of the
initial profiles, to take into account our ignorance of the supernova
progenitor properties. We show that if masses and radii of a proto-neutron star
can be determined in the first second after the birth, e.g. from gravitational
wave emission, no information could be obtained on the corresponding cold
neutron star and therefore on the cold nuclear equation of state. Similarly, it
seems unlikely that any property of the proto-neutron star equation of state
(hot and not beta-equilibrated) could be determined either, mostly due to the
lack of information on the entropy, or equivalently temperature, distribution
in such objects.
