In Brief
Retired statistician Thomas Royen solved the Gaussian correlation inequality one morning in 2014 as he brushed his teeth and posted his work online. Why did it take almost three years for it to be recognized?

Gaussian Correlation Inequality

From 1950 to 1972, the mathematical community pondered the infamous Gaussian correlation inequality (GCI). But, between 1972 and 2014, it remained unsolved, challenging talented mathematicians and statisticians for decades. Finally, in 2014, a retired German statistician had a brainstorm as he stood brushing his teeth at the sink, and solved the GCI—only no one really noticed.

The idea behind this inequality is that when two shapes overlap, the probability of hitting one of the overlapping shapes increases the chances of hitting the other. In other words, if I’m shooting paintballs at a triangle and circle that are laying one on top of one another, and my target is right in the middle, my shots will create a bell curve around that center target value, and most of my shots will exist in the overlapping areas shared by both shapes. In fact, this isn’t a random majority, but a specific, directly proportional majority controlled by the number of hits that land outside the shapes. The GCI says that the odds of hitting the overlapped area is always as high or higher than the odds of hitting inside one shape times the odds of hitting inside the other.

Everyone knew this was the case intuitively, but until 2014, no one had the proof.

Via YouTube, Technical University of Bingen
Image Credit: YouTube, Technical University of Bingen

Toothbrush Revelation

In July 2014, Thomas Royen, a retired German statistician, was brushing his teeth when he realized how to prove the GCI principle. What occurred to him, specifically, was how to calculate a key derivative; which unlocked the proof. He didn’t know how to use LaTeX, so he used Word to post his work to arXiv.org.

He sent his work to Penn State’s Donald Richards, who worked on the GCI for 30 years. Richards told Quanta that as soon as he saw Royen’s work, he knew it had been solved. It took everyone else awhile to catch on, however. Bo’az Klartag of the Weizmann Institute of Science and Tel Aviv University overlooked Royen’s proof when he received it in 2014, prompting Royen to publish it in the obscure Far East Journal of Theoretical Statistics, on whose editorial board he’d been serving.

“I am used to being frequently ignored by scientists from [top-tier] German universities,” he told Natalie Wolchover of Quanta. “I am not so talented for ‘networking’ and many contacts. I do not need these things for the quality of my life.”

However, Polish mathematician Rafał Latała and his student Dariusz Matlak could see Royen’s proof was correct, and posted their own reorganized and somewhat more detailed version to arXiv.og in 2015. This finally brought attention to Royen’s work, and by the end of 2016, the world had begun to take notice.

Perhaps almost as perplexing as the original GCI question is, why did this proof get overlooked for so long? Was it the years of false starts from other thinkers, or his somewhat modest stance in the field that caused Royen’s work to be overlooked? Today, more than ever, solutions to stubborn problems might come from anyone, anywhere. It’s important to carefully consider even answer that come from unexpected places.