  The classical Remez inequality bounds the maximum of the absolute value of a
polynomial $P(x)$ of degree $d$ on $[-1,1]$ through the maximum of its absolute
value on any subset $Z$ of positive measure in $[-1,1]$. Similarly, in several
variables the maximum of the absolute value of a polynomial $P(x)$ of degree
$d$ on the unit ball $B^n \subset {\mathbb R}^n$ can be bounded through the
maximum of its absolute value on any subset $Z\subset Q^n_1$ of positive
$n$-measure $m_n(Z)$. In \cite{Yom} a stronger version of Remez inequality was
obtained: the Lebesgue $n$-measure $m_n$ was replaced by a certain geometric
quantity $\omega_{n,d}(Z)$ satisfying $\omega_{n,d}(Z)\geq m_n(Z)$ for any
measurable $Z$. The quantity $\omega_{n,d}(Z)$ can be effectively estimated in
terms of the metric entropy of $Z$ and it may be nonzero for discrete and even
finite sets $Z$.
  In the present paper we extend Remez inequality to functions of finite
smoothness. This is done by combining the result of \cite{Yom} with the Taylor
polynomial approximation of smooth functions. As a consequence we obtain
explicit lower bounds in some examples in the Whitney problem of a $C^k$-smooth
extrapolation from a given set $Z$, in terms of the geometry of $Z$.

Classical Analysis and ODEs