A geometric approach to a class of optimization problems concerning exchangeable binary variables

In [Zaigraev, A., Kaniovski, S., 2010. Exact bounds on the probability of at least k successes in n exchangeable Bernoulli trials as a function of correlation coefficients. Statist. Probab. Lett. 80, 1079-1084] the authors present sharp bounds for the probability Rk,n of having k successes out of n exchangeable Bernoulli trials, as a function of the marginal probability of success. The result is obtained by linear programming arguments. In this paper we develop further the result utilizing a geometrical approach to the problem, and find sharp bounds for Rk,n given the marginal probability of success and the correlation among the exchangeable variables.