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We show that Bernstein polynomials are related to the Lebesgue measure on [0,1] in a manner similar as Chebyshev polynomials are related to the equilibriummeasure of [--1, 1]. We also show that Pell's polynomial equation satisfied byChebyshev polynomials, provides a partition of unity of [--1, 1], the analogueof the partition of unity of [0, 1] provided by Bernstein polynomials. Bothpartitions of unity are interpreted as a specific algebraic certificate thatthe constant polynomial ''1'' is positive-on [--1, 1] via Putinar's certificateof positivity (for Chebyshev), and-on [0, 1] via Handeman's certificate ofpositivity (for Bernstein). Then in a second step, one combines this partitionof unity with an interpretation of a duality result of Nesterov in convex conicoptimization to obtain an explicit connection with the equilibrium measure on[--1, 1] (for Chebyshev) and Lebesgue measure on [0, 1] (for Bernstein).Finally this connection is also partially established for the ''d''-dimensionalsimplex.

Chebyshev and Equilibrium Measure Vs Bernstein and Lebesgue Measure