  Interior models of giant planets traditionally assume that at a given radius
(i.e. pressure) the density should be larger than or equal to the one
corresponding to a homogeneous, adiabatic stratification throughout the planet
(referred to as the 'outer adiabat'). The observations of Jupiter's gravity
field by Juno combined with the constraints on its atmospheric composition
appear to be incompatible with such a profile. In this letter, we show that the
above assumption stems from an incorrect understanding of the
Schwarzschild-Ledoux criterion, which is only valid on a local scale. In order
to fulfil the buoyancy stability condition, the density gradient with pressure
in a non-adiabatic region must indeed rise more steeply than the {\it local}
adiabatic density gradient. However, the density gradient can be smaller than
the one corresponding to the outer adiabat at the same pressure because of the
higher temperature in an inhomogeneously stratified medium. Deep enough, the
density can therefore be lower than the one corresponding to the outer adiabat.
We show that this is permitted only if the slope of the local adiabat becomes
shallower than the slope of the outer adiabat at the same pressure, as found in
recent Jupiter models due to the increase of both specific entropy and
adiabatic index with depth. We examine the dynamical stability of this
structure and show that it is stable against non-adiabatic perturbations. The
possibility of such unconventional density profile in Jupiter complicates
further our understanding of the internal structure and evolution of
(extrasolar) giant planets.
