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 $d\ge 2$. This stands in contrast with the well-known fact that any probability distribution in ${\mathbb{R}}^d$, for any $d\ge 1$, is determined by the corresponding distributions of linear functionals. By duality, it follows that, for any $d\ge 2$, 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 ${\mathbb{R}}^d$.

4 pages; to appear in The American Mathematical Monthly. Version 2: a typo is corrected. Version 3: another typo is corrected