A Geometric Approach to Radial Correlation Type Problems

摘要

A radial probability measure is a probability measure with a density (withrespect to the Lebesgue measure) which depends only on the distances to theorigin. Consider the Euclidean space enhanced with a radial probabilitymeasure. A correlation problem concerns showing whether the radial measure ofthe intersection of two symmetric convex bodies is greater than the product ofthe radial measures of the two convex bodies. A radial measure satisfying thisproperty is said to satisfy the correlation property. A major question in thisfield is about the correlation property of the (standard) Gaussian measure. Themain result in this paper is a theorem suggesting a sufficient condition for aradial measure to satisfy the correlation property. A consequence of the maintheorem will be a proof of the correlation property of the Gaussian measure.