Abstract: Gaussian isotropic bodies and Gaussian isotropic constants were defined, and the existence and orthogonal invariability of Gaussian isotropic bodies were proved. By calculating the Gaussian isotropic constants of a ball and a cube, both with unit volume, we show that changes in the Gaussian isotropic constants in the Gaussian measure are opposite to those in the Lebesgue measure.

Key words:  convex body, Gaussian measure, isotropic constant