  In noncommutative probability theory independence can be based on free
products instead of tensor products. This yields a highly noncommutative
theory: free probability . Here we show that the classical Shannon's entropy
power inequality has a counterpart for the free analogue of entropy .
  The free entropy (introduced recently by the second named author),
consistently with Boltzmann's formula $S=k\log W$, was defined via volumes of
matricial microstates. Proving the free entropy power inequality naturally
becomes a geometric question.
  Restricting the Minkowski sum of two sets means to specify the set of pairs
of points which will be added. The relevant inequality, which holds when the
set of "addable" points is sufficiently large, differs from the Brunn-Minkowski
inequality by having the exponent $1/n$ replaced by $2/n$. Its proof uses the
rearrangement inequality of Brascamp-Lieb-L\"uttinger .
