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The extreme polygons for the self Chebyshev radius of the boundary

Evgenii V. Nikitenko, Yurii G. Nikonorov

Abstract

The paper is devoted to some extremal problems for convex polygons on the Euclidean plane, related to the concept of self Chebyshev radius for the polygon boundary. We consider a general problem of minimization of the perimeter among all $n$-gons with a fixed self Chebyshev radius of the boundary. The main result of the paper is the complete solution of the mentioned problem for $n=4$: We proved that the quadrilateral of minimum perimeter is a so called magic kite, that verified the corresponding conjecture by Rolf Walter.

The extreme polygons for the self Chebyshev radius of the boundary

Abstract

The paper is devoted to some extremal problems for convex polygons on the Euclidean plane, related to the concept of self Chebyshev radius for the polygon boundary. We consider a general problem of minimization of the perimeter among all -gons with a fixed self Chebyshev radius of the boundary. The main result of the paper is the complete solution of the mentioned problem for : We proved that the quadrilateral of minimum perimeter is a so called magic kite, that verified the corresponding conjecture by Rolf Walter.
Paper Structure (20 sections, 27 theorems, 118 equations, 22 figures)

This paper contains 20 sections, 27 theorems, 118 equations, 22 figures.

Table of Contents

  1. Introduction and the main result
  2. Some general results
  3. Some auxiliary results for boundaries of convex polygons
  4. A plan how to find extremal quadrilaterals and some preliminary considerations
  5. Case 1
  6. Case 2
  7. Case 2.1
  8. Case 2.2
  9. Case 2.3
  10. Case 3
  11. Case 3.1
  12. Case 3.2
  13. Case 3.3
  14. Case 3.4
  15. Case 4
  16. ...and 5 more sections

Key Result

Theorem 1

For any quadrilateral $P$ with the boundary $\Gamma$ on Euclidean plane, we have the inequality $L(\Gamma)\geq \frac{4}{3}\sqrt{3+2\sqrt{3}}\,\cdot \delta(\Gamma)$, with equality exactly for magic kites. In other words, a magic kite has the minimal perimeter among all quadrilaterals with a given sel

Figures (22)

  • Figure 1: A magic kite.
  • Figure 2: The transition from a non-convex quadrilateral to a convex one.
  • Figure 3: a) $N(A)$ for an interior point of $[A_i,A_{i+1}]$; b) $N(A)$ for a vertex $A=A_i$.
  • Figure 4: a) Case 2.1, b) Case 2.2, c) Case 2.3, d) Case 3.1, e) Case 3.2, f) Case 3.3, g) Case 3.4.
  • Figure 5: The ellipse and the circle for the quadrilateral $ABCD$.
  • ...and 17 more figures

Theorems & Definitions (37)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Proposition 3
  • Remark 1
  • Proposition 4
  • Proposition 5: BMNN2021
  • Proposition 6
  • Remark 2
  • ...and 27 more