  Generalizing the well-known spin-squeezing inequalities, we study the
relation between squeezing of collective $N$-particle $su(d)$ operators and
many-body entanglement geometry in multi-particle systems. For that aim, we
define the set of pseudo-separable states, which are mixtures of products of
single-particle states that lie in the $(d^2-1)$-dimensional Bloch sphere but
are not necessarily positive semidefinite. We obtain a set of necessary
conditions for states of $N$ qudits to be of the above form. Any state that
violates these conditions is entangled. We also define a corresponding
$su(d)$-squeezing parameter that can be used to detect entanglement in large
particle ensembles. Geometrically, this set of conditions defines a convex set
of points in the space of first and second moments of the collective
$N$-particle $su(d)$ operators. We prove that, in the limit $N\gg 1$, such set
is filled by pseudo-separable states, while any state corresponding to a point
outside of this set is necessarily entangled. We also study states that are
detected by these inequalities: We show that states with a bosonic symmetry are
detected if and only if the two-body reduced state violates the positive
partial transpose (PPT) criterion. On the other hand, highly mixed states
states close to the $su(d)$ singlet are detected which have a separable
two-body reduced state and are also PPT with respect to all possible
bipartitions. We also provide numerical examples of thermal equilibrium states
that are detected by our set of inequalities, comparing the spin-squeezing
inequalities with the $su(3)$-squeezing inequalities.
