Qualitative convexity and universal cross-sections

Abstract

We study a family of questions in convexity in which \emph{size does not matter}: one records a convex cross-section only up to translation and scaling, so that the data attached to a convex body $B$ and a direction $ρ$ is a path in the compact metric space $\A_{n-1}$ of \emph{aligned shapes}. The object of interest is the asymptotic behaviour of this path as the cutting hyperplane approaches the last supporting hyperplane, encoded by an invariant $T(B,ρ)$ that we call the \emph{tail}. We show that tails are always continua, that polyhedral and smooth support points are ``boring'' (the tail is a point), and that non-boring behaviour forces degenerate contact. We show that cross-section paths are locally rectifiable, that every locally rectifiable path is realisable approximately and a dense class exactly, and that exact realisation fails in general: a second-order obstruction of bounded-turning type produces a rectifiable path that is not a cross-section path. For tails, by contrast, no such restriction survives: every continuum of shapes occurs as a tail, on the nose rather than up to approximation. We construct bodies possessing \emph{nearly universal} points, at which the renormalised cross-sections approximate every planar (more generally $(n-1)$-dimensional) convex shape arbitrarily well; such points can be made dense in the boundary, with arbitrary prescribed tails at the grafting sites. Every result below has been formally verified in Lean~4. We close with several optimisation questions and a higher-codimension variant.