Does the Convex Order Between the Distributions of Linear Functionals Imply the Convex Order Between the Probability Distributions Over $\mathbb R^d$?

Abstract

It is shown that the convex order between the distributions of linear functionals does not imply the convex order between the probability distributions over $\mathbb R^d$ if $d\ge2$. This stands in contrast with the well-known fact that any probability distribution in $\mathbb R^d$, for any $d\ge1$, is determined by the corresponding distributions of linear functionals. By duality, it follows that, for any $d\ge2$, not all convex functions from $\mathbb R^d$ to $\mathbb R$ can be represented as the limits of sums $\sum_{i=1}^k g_i\circ \ell_i$ of convex functions $g_i$ of linear functionals $\ell_i$ on $\mathbb R^d$.