We review state-of-the-art optimality conditions of multivariate Chebyshev approximation, including, from oldest to newest, Kirchberger’s kernel condi- tion, Kolmogorov criteria, Rivlin and Shapiro’s annihilating measures. These conditions are then re-interpreted using the optimality conditions of convex opti- mization, subdifferential and directional derivative. Finally, this new point of view is used to derive new optimality conditions for the following problems: First for the multivariate Chebyshev approximation with a weight function. Second, the approximation problem proposed by Arzelier, Br ́ehard and Joldes (26th IEEE Symposium on Computer Arithmetic 2019) consisting in minimizing the sum of both the polynomial approximation error and the first order approximation of the worst case evaluation error of the polynomial in Horner form.