From Cauchy to Gaussian: Sharp Lower Deviations and Ehrhard-Type Inequality

Source abstract

We establish dimension-free, pointwise optimal lower-deviation bounds for convex functions under spherically symmetric Cauchy-type measures. These bounds use either mean-variance normalization or moments about the median, the latter covering the Cauchy case where the mean and variance are unavailable. By approximation with a parametrized family the Cauchy-type measures, our results recover the Gaussian setting of Paouris and Valettas \cite{PaourisValettas18} and strengthen their variance-sensitive estimate to a sharp bound, with a complete equality characterization. This goes beyond the partial improvements obtained in subsequent work. We also prove an Ehrhard-type inequality for arbitrary convex sets under these Cauchy-type measures in every dimension and characterize all equality cases. This resolves a conjecture posed by Byczkowski, Małecki and Żak \cite{ByczkowskiMaleckiZak22}. The proof follows from a variational argument combined with the joint concavity of fiber rearrangement. The geometric inequality reduces distributions of convex observables to one-dimensional convex transforms, yielding the sharp lower-deviation bounds.