  Ferroelectric materials display exotic polarization textures at the nanoscale
that could be used to improve the energetic efficiency of electronic
components. The vast majority of studies were conducted in two dimensions on
thin films, that can be further nanostructured, but very few studies address
the situation of individual isolated nanocrystals synthesized in solution,
while such structures could open other field of applications. In this work, we
experimentally and theoretically studied the polarization texture of
ferroelectric barium titanate (BaTiO$_3$, BTO) nanocrystals (NC) attached to a
conductive substrate and surrounded by air. We synthesized NC of well defined
quasi-cubic shape and 160 nm average size, that conserve the tetragonal
structure of BTO at room temperature.
  We then investigated the inverse piezoelectric properties of such pristine
individual NC by piezoresponse force microscopy (PFM), taking particular care
of suppressing electrostatic artifacts. In all the NC studied, we could not
detect any vertical PFM signal, and the maps of the lateral response all
displayed larger displacements on the edges. Using field-phase simulations
dedicated to ferroelectric nanostructures, we were able to predict the
equilibrium polarization texture. These simulations revealed that the NC core
is composed of 180{\deg} up and down domains defining the polar axis, that
rotate by 90{\deg} in the two facets orthogonal to this axis, eventually lying
within these planes forming a layer of about 10 nm thickness mainly composed of
180{\deg} domains along an edge. From this polarization distribution we
predicted the lateral PFM response, that revealed to be in very good
qualitative agreement with the experimental observations. This work positions
PFM as a relevant tool to evaluate the potential of complex ferroelectric
nanostructures to be used as sensors.
