  The theory of Chebyshev (uniform) approximation for univariate polynomial and
piecewise polynomial functions has been studied for decades. The optimality
conditions are based on the notion of alternating sequence. However, the
extension the notion of alternating sequence to the case of multivariate
functions is not trivial. The contribution of this paper is two-fold. First of
all, we give a geometrical interpretation of the necessary and sufficient
optimality condition for multivariate approximation. These optimality
conditions are not limited to the case polynomial approximation, where the
basis functions are monomials. Second, we develop an algorithm for fast
necessary optimality conditions verifications (polynomial case only). Although,
this procedure only verifies the necessity, it is much faster than the
necessary and sufficient conditions verification. This procedure is based on a
point reduction procedure and resembles the univariate alternating sequence
based optimality conditions. In the case of univariate approximation, however,
these conditions are both necessary and sufficient. Third, we propose a
procedure for necessary and sufficient optimality conditions verification that
is based on a generalisation of the notion of alternating sequence to the case
of multivariate polynomials.
