Let A(w,r) be the open ball with center w and radius r wrt the euclidean norm. Prove this set is open also wrt the max. norm (take any point X=(x_1,..,x_n) in A(w,r) and supose w = (w_1,...,w_n) ==> SUM(x_i - w_i)^2 < r^2 ==> but we also have that SUM(x_i - w_i)^2 <= n*max(x_i - w_i)^2, so if we choose wisely and s.t. |y_i - w_i| < r/Sqrt(n) then...etc. and try to end the argument by yourself. The other direction is simmilar.

Tonio