  We consider a new multivariate generalization of the classical monic
(univariate) Chebyshev polynomial that minimizes the uniform norm on the
interval $[-1,1]$. Let $\Pi^*_n$ be the subset of polynomials of degree at most
$n$ in $d$ variables, whose homogeneous part of degree $n$ has coefficients
summing up to $1$. The problem is determining a polynomial in $\Pi^*_n$ with
the smallest uniform norm on a domain $\Omega$, which we call a least Chebyshev
polynomial (associated with $\Omega$). Our main result solves the problem for
$\Omega$ belonging to a non-trivial class of domains, defined by a property of
its diagonal, and establishes the remarkable result that a least Chebyshev
polynomial can be given via the classical, univariate, Chebyshev polynomial. In
particular, the solution can be independent of the dimension. The result is
valid for fairly general domains that can be non-convex and highly irregular.
