On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes

For appropriate Gaussian processes, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-$p$ setting, as well as a version of Talagrand's result for general positive empirical processes.