A matrix form of the Brunn-Minkowski inequality

摘要

The well known Brunn-Minkowski inequality (BMI) is one of thebasic inequalities in geometry. The BMI is dual in some sense to theentropy-power inequality, which lower bounds the entropy power of thesum of independent random variables. The authors derive a matrix formfor the BMI and discuss its applications. They note that the matrix BMIcan 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 ofcodewords of lattice constellations or lattice quantizers satisfyingspectral constraints