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In 'Tractable semi-algebraic approximation using Christoffel-Darboux kernel'Marx, Pauwels, Weisser, Henrion and Lasserre conjectured, that theapproximation rate $\mathcal O(\frac 1 {\sqrt(d)})$ of a Lipschitz functions bya semi-algebraic function induced by a Christoffel- Darboux kernel of degree$d$ in the $L^1$ norm can be improved for more regular functions. Here we willshow, that for semi-algebraic and definable functions the results can bestrengthened to a rational approximation rate in the $L^\infty$ norm.

Some suggestions concerning the conjecture in: 'Tractable semi-algebraic approximation using Christoffel-Darboux kernel'