The Projection Gap

A probability distribution in R^d is uniquely determined by its one-dimensional projections. If you know the distribution of every linear functional of a random vector — every possible one-dimensional shadow — you can reconstruct the full d-dimensional distribution exactly. This is the Cramér-Wold theorem, and it’s one of the most useful facts in multivariate statistics.

The natural extension would be: if distribution A dominates distribution B in convex order on every one-dimensional projection, then A dominates B in convex order in the full space. Convex order is the comparison that says “A is more spread out than B” — formally, E[f(X)] ≥ E[f(Y)] for every convex function f.

The result (arXiv:2510.04269) is a clean counterexample: for d ≥ 2, convex order on all projections does not imply convex order in the original space. Two distributions can satisfy A ≥ B in convex order on every one-dimensional shadow simultaneously, yet A does not dominate B in convex order in R^d.

The duality version is equally sharp: not all convex functions from R^d to R can be represented as limits of sums of convex functions of linear functionals. The class of “projection-decomposable” convex functions is strictly smaller than the class of all convex functions, starting in dimension 2.

The structural insight: determination and ordering obey different logics. Projections carry enough information to identify a distribution uniquely (the reconstruction problem), but not enough to preserve its order relationships (the comparison problem). Identification requires only that different distributions produce different projections somewhere. Ordering requires that the comparison hold everywhere simultaneously — a much stronger condition that projections cannot enforce because the convex functions that define the order in R^d are richer than those constructible from one-dimensional convex functions.

“Does the Convex Order Between the Distributions of Linear Functionals Imply the Convex Order Between the Probability Distributions Over R^d?” arXiv:2510.04269 (2025).