Approximation characteristics for diagonal operators in different computational settings

First we study several extremal problems on minimax, and prove that they are equivalent. Then we connect this result with the exact values of some approximation characteristics of diagonal operators in different settings, such as the best n-term approximation, the linear average and stochastic n-widths, and the Kolmogorov and linear n-widths. Most of these exact values were known before, but in terms of equivalence of these extremal problems, we present a unified approach to give them a direct proof.