Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hyperbolic width functions and characterizations of bodies of constant width in the hyperbolic space

Károly J. Böröczky, András Csépai, Ádám Sagmeister

Abstract

We discuss basic properties of several different width functions in the $n$-dimensional hyperbolic space such as continuity, and we also define a new hyperbolic width as the extension of Leichtweiss' width function. Then we prove a characterization theorem of bodies of constant width regarding the aforementioned notions of hyperbolic width.

Hyperbolic width functions and characterizations of bodies of constant width in the hyperbolic space

Abstract

We discuss basic properties of several different width functions in the -dimensional hyperbolic space such as continuity, and we also define a new hyperbolic width as the extension of Leichtweiss' width function. Then we prove a characterization theorem of bodies of constant width regarding the aforementioned notions of hyperbolic width.
Paper Structure (16 sections, 37 theorems, 71 equations, 4 figures)

This paper contains 16 sections, 37 theorems, 71 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Models of the hyperbolic space
  3. The hyperboloid model
  4. The Poincaré ball model
  5. The Beltrami--Cayley--Klein model
  6. Compact subsets in the hyperbolic space
  7. Hyperbolic width functions
  8. Santaló's width function
  9. Fillmore's width function
  10. Leichtweiss' width function
  11. Planar alternatives
  12. A new concept of width
  13. Continuity of width functions
  14. Width and diameter
  15. Bodies of constant width in the hyperbolic space
  16. ...and 1 more sections

Key Result

Theorem 1.1

The width functions $w_S(K,(z,H_z)$, $w_F(K,i)$, $w_L(K,(p,H_p))$, $w_{JCJL}(K,(z,\ell_z,z'))$, $w_{GH}(K,i)$ and $w(K,H)$ are continuous in all of their parameters. The minimal widths $w_S(K)$, $w_F(K)$, $w_L(K,p)$, $w_{JCJL}(K)$, $w_{GH}(K)$ and $w(K)$ of convex bodies $K$ are also continuous in $

Figures (4)

  • Figure 1: The Santaló width $w_S\left(K,\left(z,H_z\right)\right)$ of a regular triangle $K$ at a boundary point $z$ and supporting line $H_z$ at $z$
  • Figure 2: The minimal Santaló width is not monotonic
  • Figure 3: The Fillmore width $w_F\left(K,i\right)$ of a regular triangle $K$ at the ideal point $i$
  • Figure 4: The Leichtweiss width $w_L\left(K,\left(p,H\right)\right)$ of a regular triangle $K$ at a line $H$ that contains the inner point $p$

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • Corollary 3.2
  • Lemma 3.3
  • Theorem 3.4: Blaschke
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 27 more