The classical Remez inequality [‘Sur une propriété des polynomes de Tchebycheff’, Comm. Inst. Sci. Kharkov13 (1936), 9–95] 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 (see, for example, Brudnyi and Ganzburg [‘On an extremal problem for polynomials of
$n$
variables’, Math. USSR Izv.37 (1973), 344–355], Yomdin [‘Remez-type inequality for discrete sets’, Israel. J. Math.186 (2011), 45–60], Brudnyi [‘On covering numbers of sublevel sets of analytic functions’, J. Approx. Theory162 (2010), 72–93]). 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
$\mathbb{R}^{n}$
), that is, satisfies a Remez-type inequality:
$\sup _{[-1,1]^{n}}|P|\leq C\sup _{Z}|P|$
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 precisely 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 nontrivial examples. Next, we extend the Turán–Nazarov inequality for exponential polynomials to several variables, and on this basis 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.