Convex Order Beyond Dimension One: Projection Tests, Counterexamples and Gaussian Mixtures

In actuarial science and quantitative finance, convex order provides a natural way to compare risks with the same mean. In dimension one, convex order is well understood through several characterisations. In higher dimensions, a natural approach is to compare all one-dimensional projections, but, although necessary, the resulting condition is in general not sufficient for multivariate convex order. A simple counterexample due to Pinelis exploits Popoviciu's inequality. We revisit this example using gauge functions, and show that gauge functions themselves do not suffice to characterise multivariate convex order for symmetric distributions, even in dimension two. We then show that the radial case makes the source of the failure transparent and construct radial counterexamples in every dimension at least two. Nevertheless, projection tests are sufficient for some important families of distributions. In particular, this is the case for linear transformations of a common radially distributed random vector. This framework includes elliptical distributions generated from a common radial distribution and, in particular, centred Gaussian distributions. Beyond comparisons between Gaussian distributions, Jourdain and Pag\`es recently studied the comparison of a Gaussian distribution with a Gaussian mixture, obtaining necessary conditions and sufficient conditions whose equivalence was left open in general. In this paper, we settle the comparison of a centred Gaussian distribution with a mixture of two centred Gaussian distributions: in every finite dimension, the Gaussian is dominated by the mixture in multivariate convex order if and only if all one-dimensional projections are ordered in convex order.