The Gaussian isoperimetric inequality and decoding error probabilities for the Gaussian channel

The Gaussian isoperimetric inequality states that among all sets in Rⁿ with prescribed Gaussian measure, the half-spaces have minimal Gaussian perimeter. We apply this result to Voronoi regions of codes in Euclidean space and obtain a surprisingly precise description of how the maximum-likelihood decoding error probability varies as a function of the minimum Euclidean distance.