Subgaussian Convex Bodies Need Not Have Subgaussian Projections

Every centrally symmetric convex body is an orthogonal projection of a centrally symmetric convex body whose linear functionals have psi_2/L_2 ratio at most 2 sqrt(2), independently of dimension. The lift is the intersection of a Euclidean ball with a cylinder in a sufficiently large dimension. In particular, for every n >= 2 we give an explicit body C_n in R^(64n^2) with this bound whose n-dimensional coordinate projection is (1/2) B_1^n. The latter has psi_2/L_2 ratio at least sqrt(n+2)/(2 sqrt(2 log 2)). Thus dimension-free subgaussianity of uniform volume does not survive geometric projection. This gives a negative answer to the projection subquestion in Paouris's Problem 7 in the AIM list on algorithmic convex geometry. It does not address the zonoid subquestion. The bad directional constant of B_1^n is known prior work and is credited. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.