On uniform constants of strong uniqueness in Chebyshev approximations and fundamental results of N. G. Chebotarev

In the problem of the best uniform approximation of a continuous real-valued function $f\in C(Q)$ in a finite-dimensional Chebyshev subspace $M\subset C(Q)$, where $Q$ is a compactum, one studies the positivity of the uniform strong uniqueness constant $\gamma(N)=\inf\{\gamma(f)\colon f\in N\}$. Here $\gamma(f)$ stands for the strong uniqueness constant of an element $f_M\in M$ of best approximation of $f$, that is, the largest constant $\gamma>0$ such that the strong uniqueness inequality $\|f-\varphi\|\geqslant\|f-f_M\|+\gamma\|f_M-\varphi\|$ holds for any $\varphi\in M$. We obtain a characterization of the subsets $N\subset C(Q)$ for which there is a neighbourhood $O(N)$ of $N$ satisfying the condition $\gamma(O(N))>0$. The pioneering results of N. G. Chebotarev were published in 1943 and concerned the sharpness of the minimum in minimax problems and the strong uniqueness of algebraic polynomials of best approximation. They seem to have been neglected by the specialists, and we discuss them in detail.