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 An−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.
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BibTeX
@misc{feldman2026qualitative,
title = {{Qualitative convexity and universal cross-sections}},
author = {David Victor Feldman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26129},
eprint = {2609.26129},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26129},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Qualitative convexity and universal cross-sections
AU - David Victor Feldman
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26129
UR - https://arxiv.org/abs/2609.26129
AB - 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.
ER -