A matrix form of the Brunn-Minkowski inequality

The well known Brunn-Minkowski inequality (BMI) is one of the basic inequalities in geometry. The BMI is dual in some sense to the entropy-power inequality, which lower bounds the entropy power of the sum of independent random variables. The authors derive a matrix form for the BMI and discuss its applications. They note that the matrix BMI can be used to lower bound the volume of a projection of a lattice cell, and so it can find applications in calculating the effective number of codewords of lattice constellations or lattice quantizers satisfying spectral constraints.