On polynomials of the best approximation in the Hausdorff metric

A definition of the Hausdorff alternance is given. In these terms we obtain a sufficient condition for an algebraic polynomial to have minimal deviation from the function $f$ in the Hausdorff $\alpha$-metric. A condition under which a polynomial $P_n$ is the unique polynomial of best approximation to a function $f$, as well as a necessary condition for $P_n$ to have minimal deviation from $f$ are established. Also, similar theorems for $2\pi$-periodic functions are stated. Bibliography: 3 titles.