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An in-depth study of ball-bodies

Shiri Artstein-Avidan, Dan I. Florentin

TL;DR

This work introduces and analyzes ball-bodies ${\cal S}_n$, defined as intersections of translates of the unit ball, and studies the associated $c$-duality $A^c=\bigcap_{x\in A}B(x,1)$. It establishes that $K\in{\cal S}_n$ iff $K^{cc}=K$, that $K^c$ is the unique, isometric dual in this class, and that c-duality is essentially the only nontrivial order-reversing involution on ${\cal S}_n$ up to rigid motions. The paper develops a rich iso-parametric theory, deriving Santaló-type and Mahler-type inequalities, precise in-/out-radius relations, and diameter bounds, and it analyzes the boundary structure via $c$-extremality and curvature, including Carathéodory-type theorems. It further investigates symmetrizations and shadow systems, proving closure under Minkowski averages but showing limitations of Steiner symmetrization (preserved in the plane but not in dimension 3), and it connects these geometric insights to measure transport, Kneser–Poulsen-type phenomena, and constant-width bodies, with detailed results in the plane and informative higher-dimensional examples. The work closes with open problems, a detailed appendix of special $c$-class bodies, and applications to lens intersections and constant-width basins, illustrating the central role of the $c$-duality framework in modern convex-geometric questions.

Abstract

In this paper we study the class of so called `ball-bodies' in ${\mathbb R}^n$, given by intersections of translates of Euclidean unit balls (or, equivalently, summand of the Euclidean ball). We study the class along with the natural duality operator defined on it. The class is naturally linked to many interesting problems in convex geometry, including bodies of constant width and the Knesser-Poulsen conjecture. We discuss old and new inequalities of isoperimetric type and of Santaló type, in this class. We study the boundary structure of bodies in the class, Carathéodory type theorem and curvature relations. We discuss various symmetrizations with relation to this class, and make some first steps regarding problems for bodies of constant width.

An in-depth study of ball-bodies

TL;DR

This work introduces and analyzes ball-bodies , defined as intersections of translates of the unit ball, and studies the associated -duality . It establishes that iff , that is the unique, isometric dual in this class, and that c-duality is essentially the only nontrivial order-reversing involution on up to rigid motions. The paper develops a rich iso-parametric theory, deriving Santaló-type and Mahler-type inequalities, precise in-/out-radius relations, and diameter bounds, and it analyzes the boundary structure via -extremality and curvature, including Carathéodory-type theorems. It further investigates symmetrizations and shadow systems, proving closure under Minkowski averages but showing limitations of Steiner symmetrization (preserved in the plane but not in dimension 3), and it connects these geometric insights to measure transport, Kneser–Poulsen-type phenomena, and constant-width bodies, with detailed results in the plane and informative higher-dimensional examples. The work closes with open problems, a detailed appendix of special -class bodies, and applications to lens intersections and constant-width basins, illustrating the central role of the -duality framework in modern convex-geometric questions.

Abstract

In this paper we study the class of so called `ball-bodies' in , given by intersections of translates of Euclidean unit balls (or, equivalently, summand of the Euclidean ball). We study the class along with the natural duality operator defined on it. The class is naturally linked to many interesting problems in convex geometry, including bodies of constant width and the Knesser-Poulsen conjecture. We discuss old and new inequalities of isoperimetric type and of Santaló type, in this class. We study the boundary structure of bodies in the class, Carathéodory type theorem and curvature relations. We discuss various symmetrizations with relation to this class, and make some first steps regarding problems for bodies of constant width.
Paper Structure (34 sections, 94 theorems, 174 equations, 2 figures)

This paper contains 34 sections, 94 theorems, 174 equations, 2 figures.

Table of Contents

  1. Definitions and First Observations
  2. Continuity, Isometry and Uniqueness
  3. Characterization of the $c$-duality on ${\cal S}^n$
  4. Continuity properties of the $c$-duality
  5. Approximation
  6. Iso-parametric inequalities
  7. In-radius, Out-radius and Diameter
  8. Contact points of a body and its in/out-ball
  9. Fixing $r(K)$ and extremizing $R(K)$ and ${\rm diam}(K)$
  10. Extremizing volume for given $r(K)$ or $R(K)$
  11. The intersection of $K$ and $K^c$
  12. $c$-Mahler in the plane
  13. Boundary Structure
  14. Extremal Points
  15. Duality and $c$-extremality
  16. ...and 19 more sections

Key Result

Proposition 1.5

Let $K\subset \mathbb R^n$, then $K\in {\cal S}_n$ if and only if for any $x_0, x_1\in K$ we have $\{x_0, x_1\}^{cc} \subseteq K$.

Figures (2)

  • Figure 1: The set in Example \ref{['ex:Naztel-body']}
  • Figure 2: The $c$-dual of $E$, for $b=1/2$

Theorems & Definitions (213)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Proposition 1.5
  • proof : Proof of Proposition \ref{['prop:ball and spindle convexity']}
  • Proposition 1.6
  • proof
  • Remark 1.7
  • Lemma 1.8
  • ...and 203 more