Abstract: The aim of the talk is to present a Remez-type inequality for sets with cusps. Recall the classical Remez inequality: Suppose that V ⊂ [0; 1] is measurable and |V| > 0. Then, for each P ∈ R[X] with deg P≤n, ||P||_{[0; 1]} ≤ T_n ((2 - |V|) / |V|) ||P||_V, where |V| denotes the Lebesgue measure of V, and T_n is the Chebyshev polynomial of degree n. There is a rich literature on the subject, including various generalizations of Remez's result. However, the available papers deal with mostly univariate or (multivariate) convex case.
The problem of Remez-type inequality in dimensions higher than one (that is, if we replace the interval [0; 1] by a multidimensional set) seems to be difficult. One can expect that for convex sets it should be possible to reduce somehow the problem to dimension one. And it is the case - a version of Remez inequality for convex sets is due to Brudnyi and Ganzburg. The situation is completely different if we consider nonconvex sets - it is not even clear how to tackle the sets that have "tame" topology.