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In 1889 A. Markov proved that for every polynomial p in one variable the inequality $\left|\right|p^{\text{'}}{\left|\right|}_{[-1,1]}{\le \left(degp\right)²\left|\right|p\left|\right|}_{[-1,1]}$ is true. Moreover, the exponent 2 in this inequality is the best possible one. A tangential Markov inequality is a generalization of the Markov inequality to tangential derivatives of certain sets in higher-dimensional Euclidean spaces. We give some motivational examples of sets that admit the tangential Markov inequality with the sharp exponent. The main theorems show that the results on certain arcs and surfaces,...

We give another proof of the fact that any semialbraic curve admits a tangential Markov inequality. We establish this inequality on semialgebraic surfaces with finitely many singular points.

It is known that for $C^{\infty }$ determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not $C^{\infty }$ determining. In this paper we give examples of sets which are not $C^{\infty }$ determining, but have the Bernstein and generalized Markov properties.