Isoperimetric and Stable Sets for Log-ConcavePerturbations of Gaussian Measures

• DOI: 10.2478/agms-2014-0014
• Publication Date: 2014-12-12T16:02:22Z
• Authors: César Rosales
• Keywords: Hyperplane; Perimeter; Gaussian measure

Abstract Let be an open half-space or slab in ℝ n+1 endowed with a perturbation of the Gaussian measureof form f (p) := exp(ω(p) − c|p| 2 ), where c > 0 and ω is smooth concave function depending only onthe signed distance from linear hyperplane parallel to ∂ Ω. In this work we follow variational approach show that half-spaces perpendicular Ω uniquely minimize weighted perimeter among sets enclosing same volume. The main ingredient proof characterization half-spacesparallel as unique stable small singular set null capacity.Our methods also apply for = , which produces particular classification Gaussspace new isoperimetric inequality. Finally, use optimal transport studythe minimizers when term possibly non-smooth.