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A full classification of the isometries of the class of ball-bodies

Shiri Artstein-Avidan, Arnon Chor, Dan Florentin

Abstract

Complementing our previous results, we give a classification of all isometries (not necessarily surjective) of the metric space consisting of ball-bodies, endowed with the Hausdorff metric. "Ball bodies" are convex bodies which are intersections of translates of the Euclidean unit ball. We show that any such isometry is either a rigid motion, or a rigid motion composed with the c-duality mapping. In particular, any isometry on this metric space has to be surjective.

A full classification of the isometries of the class of ball-bodies

Abstract

Complementing our previous results, we give a classification of all isometries (not necessarily surjective) of the metric space consisting of ball-bodies, endowed with the Hausdorff metric. "Ball bodies" are convex bodies which are intersections of translates of the Euclidean unit ball. We show that any such isometry is either a rigid motion, or a rigid motion composed with the c-duality mapping. In particular, any isometry on this metric space has to be surjective.

Paper Structure

This paper contains 6 theorems, 20 equations.

Key Result

Theorem 1

Let $T:{\mathcal{S}}_n\rightarrow {\mathcal{S}}_n$ be an isometry (not assumed a-priori to be surjective) with respect to the Hausdorff distance. Then there exists a rigid motion (orthogonal transformation and translation) $g:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that either $TK = gK$ for all

Theorems & Definitions (12)

  • Theorem 1
  • Definition 2
  • Lemma 3
  • Proposition 4
  • proof : Proof of Proposition \ref{['prop:point_in_image']}
  • Lemma 5
  • proof
  • Lemma 6
  • proof : Proof of Theorem \ref{['thm:main_not_bijective2']}
  • Proposition 8
  • ...and 2 more