Estimating positive processes

Recently, the Talagrand conjecture regarding selector processes has been proved: work by Pham and Park. This conjecture concerned the characterization of the expected value of the process supremum in terms of the existence of an appropriate small coverage of a random event describing the process supremum exceeding a multiple of its mean. Together with Rafał Martynek and Rafał Meller, we developed a clear proof of this result and then dealt with its generalizations. The most important extension for us was to find an analogous characterization of the expected value for positive canonical processes. It turns out that such a characterization exists - our team is currently finishing the work with this result.