For fixed $w$ and $m,m^{\prime}$ satisfying $\langle m,w\rangle=\langle m^{\prime},w\rangle=1$, the convexity of $u_{m}$ and $u_{m^{\prime}}$ on the $w$-chart are equivalent because $m-m^{\prime}$ is an affine function. However, on the overlap of the $w$-chart and the $w^{\prime}$-chart, if $u_{m}$ is convex in one chart it is not automatically convex in the other.