  In this paper, we derive optimality conditions (Chebyshev approximation) for
multivariate functions. The theory of Chebyshev (uniform) approximation for
univariate functions is very elegant. The optimality conditions are based on
the notion of alternance (maximal deviation points with alternating deviation
signs). It is not very straightforward, however, how to extend the notion of
alternance to the case of multivariate functions. There have been several
attempts to extend the theory of Chebyshev approximation to the case of
multivariate functions. We propose an alternative approach, which is based on
the notion of convexity and nonsmooth analysis.
