Apparently, Bednorz and Latała have solved Talagrand’s $5,000 Bernoulli Conjecture. The conjecture concerns the supremum of a Bernoulli process.
Consider a finite subset and define the value
where are i.i.d. random . This looks somewhat similar to the corresponding value
where are i.i.d. standard normal random variables. But while can be characterized (up to universal constant factors) by the Fernique-Talagrand majorizing measures theory, no similar control was known for . One stark difference between the two cases is that depends only on the distance geometry of , i.e. whenever is an affine isometry. On the other hand, can depend heavily on the coordinate structure of .
This can be proved easily using Jensen’s inequality.
Talagrand’s Bernoulli conjecture is that can be characterized by these two upper bounds.
Bernoulli conjecture: There exists a constant such that for every , there are two subsets such that
The set controls the “Gaussian” part of the Bernoulli process, while the set controls the part that is heavily dependent on the coordinate structure.
This beautiful problem finally appears to have met a solution. While I don’t know of any applications yet in TCS, it does feel like something powerful and relevant.