  The classical Remez inequality bounds the maximum of the absolute value of a
real polynomial $P$ of degree $d$ on $[-1,1]$ through the maximum of its
absolute value on any subset $Z\subset [-1,1]$ of positive Lebesgue measure.
Extensions to several variables and to certain sets of Lebesgue measure zero,
massive in a much weaker sense, are available.
  Still, given a subset $Z\subset [-1,1]^n\subset {\mathbb R}^n$ it is not easy
to determine whether it is ${\mathcal P}_d({\mathbb R}^n)$-norming (here
${\mathcal P}_d({\mathbb R}^n)$ is the space of real polynomials of degree at
most $d$ on ${\mathbb R}^n$), i.e. satisfies a Remez-type inequality:
$\sup_{[-1,1]^n}|P|\le C\sup_{Z}|P|$ for all $P\in {\mathcal P}_d({\mathbb
R}^n)$ with $C$ independent of $P$. (Although ${\mathcal P}_d({\mathbb
R}^n)$-norming sets are exactly those not contained in any algebraic
hypersurface of degree $d$ in ${\mathbb R}^n$, there are many apparently
unrelated reasons for $Z \subset [-1,1]^n$ to have this property.)
  In the present paper we study norming sets and related Remez-type
inequalities in a general setting of finite-dimensional linear spaces $V$ of
continuous functions on $[-1,1]^n$, remaining in most of the examples in the
classical framework. First, we discuss some sufficient conditions for $Z$ to be
$V$-norming, partly known, partly new, restricting ourselves to the simplest
non-trivial examples. Next, we extend the Turan-Nazarov inequality for
exponential polynomials to several variables, and on this base prove a new
fewnomial Remez-type inequality. Finally, we study the family of optimal
constants $N_{V}(Z)$ in the Remez-type inequalities for $V$, as the function of
the set $Z$, showing that it is Lipschitz in the Hausdorff metric.
