Gaussian correlation inequality

The Gaussian correlation inequality (GCI), formerly known as the Gaussian correlation conjecture (GCC), is a mathematical theorem in the fields of mathematical statistics and convex geometry. A special case of the inequality was published as a conjecture in a paper from 1955; further development was given by Olive Jean Dunn in 1958; and the general case was stated in 1972, also as a conjecture.

The inequality remained unproved until 2014, when Thomas Royen, a German statistician, proved it using relatively elementary tools. The proof was not generally known when it was published in 2014, due to Royen's relative anonymity and his choice to publish the proof in a predatory journal. Another reason was multiple futile attempts to prove it, causing skepticism among mathematicians in the field.

The conjecture, and its solution, came to public attention in 2017, when reports of Royen's proof were published in mainstream media.

Let ${\displaystyle \mu _{n}}$ be an n-dimensional Gaussian measure on ${\displaystyle \mathbb {R} ^{n}}$ centered at the origin. Then for all convex sets ${\displaystyle E,F\subset \mathbb {R} ^{n}}$ that are symmetric about the origin,

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