Algebraic polynomials least deviating from zero in measure on a segment

We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment –1, 1] with respect to a measure, or, more precisely, with respect to the functional μ(f) = mes{x ∈ –1, 1]: ∣f (x)∣ ≥ 1}. We also discuss an analogous problem with respect to the integral functionals ∫_–1¹ φ (∣f (x)∣) dx for functions φ that are defined, nonnegative, and nondecreasing on the semiaxis 0, +∞).