Does the Convex Order Between the Distributions of Linear Functionals Imply the Convex Order Between the Probability Distributions Over ℝ <i> ^d </i> ?

It is shown that the convex order between the distributions of linear functionals does not imply the convex order between the probability distributions over (Formula presented.) if (Formula presented.). This stands in contrast with the well-known fact that any probability distribution in (Formula presented.), for any (Formula presented.), is determined by the corresponding distributions of linear functionals. By duality, it follows that, for any (Formula presented.), not all convex functions from (Formula presented.) to (Formula presented.) can be represented as the limits of sums (Formula presented.) of convex functions (Formula presented.) of linear functionals (Formula presented.) on (Formula presented.).