Gaussian correlation inequality

If we now consider a circle and a rectangle in the plane, both centered at the origin, then the proportion of the darts landing in the intersection of both shapes is no less than the product of the proportions of the darts landing in each shape.

A special case of the inequality was conjectured in 1955;[1] further development was given by Olive Jean Dunn in 1958.

[6] The general case of the inequality remained open until 2014, when Thomas Royen, a retired German statistician, proved it using relatively elementary tools.

[7] In fact, Royen generalized the conjecture and proved it for multivariate gamma distributions.

[2][8] Another reason was a history of false proofs (by others) and many failed attempts to prove the conjecture, causing skepticism among mathematicians in the field.

The gaussian correlation inequality states that probability of hitting both circle and rectangle with a dart is greater than or equal to the product of the individual probabilities of hitting the circle or the rectangle.