For -(n+1)<p<-1, isotropic Lp Gaussian Minkowski solutions with R(K)≤1 are unique and spherical, without assuming the body is origin-centered.
Sharp Poincar\'e-type inequality for the Gaussian measure on the boundary of convex sets
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abstract
A sharp Poincar\'e-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso second-variation inequality. The new inequality is nothing but an infinitesimal form of Ehrhard's inequality for the Gaussian measure.
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Uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem
For -(n+1)<p<-1, isotropic Lp Gaussian Minkowski solutions with R(K)≤1 are unique and spherical, without assuming the body is origin-centered.