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Homothetic packings of centrally symmetric convex bodies

Sean Dewar

TL;DR

This work generalizes rigidity and realizability results from disc packings to homothetic packings of regular symmetric convex bodies in the plane. It develops a framework of packing rigidity maps in normed spaces, connecting geometric packing constraints with graph-theoretic sparsity conditions. The main contributions show that for a regular symmetric body $C$ not a disc transform, almost all radii yield $(2,2)$-sparse contact graphs, with independence and sticky rigidity characterized by rank conditions; additionally, a comeagre set of $C$-bodies ensures that every $(2,2)$-sparse planar graph can be realized as an independent packing. These results extend classic planar packing realizability (Schramm, KAT) to a broad, generic class of norms and illuminate how geometric properties of $C$ influence rigidity and graph realizability in packings.

Abstract

A centrally symmetric convex body is a convex compact set with non-empty interior that is symmetric about the origin. Of particular interest are those that are both smooth and strictly convex -- known here as regular symmetric bodies -- since they retain many of the useful properties of the $d$-dimensional Euclidean ball. We prove that for any given regular symmetric body $C$, a homothetic packing of copies of $C$ with randomly chosen radii will have a $(2,2)$-sparse planar contact graph. We further prove that there exists a comeagre set of centrally symmetric convex bodies $C$ where any $(2,2)$-sparse planar graph can be realised as the contact graph of a stress-free homothetic packing of $C$.

Homothetic packings of centrally symmetric convex bodies

TL;DR

This work generalizes rigidity and realizability results from disc packings to homothetic packings of regular symmetric convex bodies in the plane. It develops a framework of packing rigidity maps in normed spaces, connecting geometric packing constraints with graph-theoretic sparsity conditions. The main contributions show that for a regular symmetric body not a disc transform, almost all radii yield -sparse contact graphs, with independence and sticky rigidity characterized by rank conditions; additionally, a comeagre set of -bodies ensures that every -sparse planar graph can be realized as an independent packing. These results extend classic planar packing realizability (Schramm, KAT) to a broad, generic class of norms and illuminate how geometric properties of influence rigidity and graph realizability in packings.

Abstract

A centrally symmetric convex body is a convex compact set with non-empty interior that is symmetric about the origin. Of particular interest are those that are both smooth and strictly convex -- known here as regular symmetric bodies -- since they retain many of the useful properties of the -dimensional Euclidean ball. We prove that for any given regular symmetric body , a homothetic packing of copies of with randomly chosen radii will have a -sparse planar contact graph. We further prove that there exists a comeagre set of centrally symmetric convex bodies where any -sparse planar graph can be realised as the contact graph of a stress-free homothetic packing of .

Paper Structure

This paper contains 7 sections, 10 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normed spaces and convex bodies
  4. Rigidity of frameworks in normed spaces
  5. Sticky rigidity for packings with random radii
  6. Centrally symmetric convex body packings
  7. Packing rigidity maps

Key Result

Theorem 1.1

schramm90 Let $C$ be a smooth convex body in the plane and $G$ a planar graph. Then there exists a $C$-packing $P$ with contact graph $G$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.1
  • ...and 5 more