Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality

Details Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality

1995-10-27

arxiv.org/abs/math/9510203v1

Mathematics

Probability

Scientific paper

10.1007/BF02108815

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 .

Affiliated with

Also associated with

No associations

Rating

Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-729359