Consider the Chebyshev distance in two dimensions: $ C[x,y] := \max\left(\text{abs}(x-x_0),\text{abs}(y-y_0)\right) $
Is $C[x,y]$ a convex function of $(x,y)$? Now I think, say $\frac{dC[x,y]}{dx}$ is not smooth so I don't think we can use the condition $f^{''}>0$ to prove the convexity of the Chebyshev distance.