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The main source of Chebyshev subspaces is provided by divisor spaces occurring in algebraic geometry. Let D be a divisor (a formal finite sum of points with integer multiplicities) in the Riemann sphere that is symmetric relative to the real axis and assume that D C n 1 0. By the space of this divisor we shall mean the subspace of polynomials in (2) such that the multiplicities of their zeros (and poles: a pole has a negative multiplicity) at an arbitrary point in the Riemann sphere are no smaller than the multiplicity of this point in the divisor: L .

Prove the second sD1 theorem of V. A. x/i D 0, and (2) some quantities in the set . xs /, s D 1; : : : ; m, are distinct from zero, but no two of them have distinct signs. Hint. x/ mentioned in the first Markov theorem. x/. xs / is opposite to the sign of . 1/s we can derive a contradiction with conditions (1) and (2). 6. Prove a theorem due to Carath´eodory: let X be a subset of Rn ; then any point in its convex hull is a convex combination of at most n C 1 points in X . 7. T /, provided that m Ä n C 1.

1. x/ is a (not necessarily unique) solution of V. A. x/ jP i is a solution of Problem A with one constraint hp jP i D 1. x/=kP kE is a solution of V. A. Markov’s problem. 5 Other Applications The following problems provide further examples of problems of least deviation in the uniform metric: 1. Picking interpolation nodes for functions of prescribed smoothness defined on a compact subset of the real axis . 2. n C 1 kC k /-alternance on Œ 1; 1. 3. Optimizing one-dimensional quadrature formulae of Gauss type [31, 97].