  Relations among von Neumann entropies of different parts of an $N$-partite
quantum system have direct impact on our understanding of diverse situations
ranging from spin systems to quantum coding theory and black holes. Best
formulated in terms of the set $\Sigma^*_N$ of possible vectors comprising the
entropies of the whole and its parts, the famous strong subaddivity inequality
constrains its closure $\overline\Sigma^*_N$, which is a convex cone. Further
homogeneous constrained inequalities are also known.
  In this work we provide (non-homogeneous) inequalities that constrain
$\Sigma_N^*$ near the apex (the vector of zero entropies) of
$\overline\Sigma^*_N$, in particular showing that $\Sigma_N^*$ is not a cone
for $N\geq 3$. Our inequalities apply to vectors with certain entropy
constraints saturated and, in particular, they show that while it is always
possible to up-scale an entropy vector to arbitrary integer multiples it is not
always possible to down-scale it to arbitrarily small size, thus answering a
question posed by A. Winter. Relations of our work to topological materials,
entanglement theory, and quantum cryptography are discussed.
