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On the maximum product of distances of diameter $2$ point sets

Stijn Cambie, Arne Decadt, Yanni Dong, Tao Hu, Quanyu Tang

Abstract

We consider a problem posed by Erdős, Herzog and Piranian on the maximum product of distances of a point set of order $n$ with a given diameter. We prove that it is sufficient to consider convex polygons and obtain results on the structure of the diameter graph. We also give constructions that drastically improve on the regular $n$-gons, sketching what the extremal polygons should look like, while presenting results indicating that one cannot hope to characterize the extremal polygons in general for even orders.

On the maximum product of distances of diameter $2$ point sets

Abstract

We consider a problem posed by Erdős, Herzog and Piranian on the maximum product of distances of a point set of order with a given diameter. We prove that it is sufficient to consider convex polygons and obtain results on the structure of the diameter graph. We also give constructions that drastically improve on the regular -gons, sketching what the extremal polygons should look like, while presenting results indicating that one cannot hope to characterize the extremal polygons in general for even orders.
Paper Structure (29 sections, 31 theorems, 235 equations, 6 figures)

This paper contains 29 sections, 31 theorems, 235 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Normalization and benchmark configurations
  4. Background and challenges
  5. Main contributions
  6. Organization and open conjectures
  7. Asymptotic notation
  8. Declaration of AI usage
  9. On the structure of extremal configurations
  10. Extremal constructions for small $n$
  11. Up to $4$ points
  12. $n \in \{5,6\}$
  13. The extremal octagon and decagon ($n \in \{8,10\}$)
  14. The extremal dodecagon
  15. Values obtained for more small values
  16. ...and 14 more sections

Key Result

Theorem 1

Let $P=\{z_1,\dots,z_n\}\subset\mathbb C$ satisfy $\mathop{\mathrm{diam}}\nolimits(P)\leqslant 2$ and attain $\Delta(P)=\Delta_{\max}(n)$. Then the diameter graph on vertex set $P$ is either unicyclic or a caterpillar.

Figures (6)

  • Figure 1: Construction for $n=4$.
  • Figure 2: All diameter graphs for $n=6$ allowed by \ref{['thm:Woodall']}.
  • Figure 3: Illustration of the extremal construction for $n=8$ and $n=10$
  • Figure 4: Constructions for $n=6$ and $n=12$ assuming dihedral symmetry and 3 pendant edges along the symmetry axes.
  • Figure 5: The shape of the polygon $B_k B_{2k} B_{3k} B_{4k} B_{5k} B_{6k}$.
  • ...and 1 more figures

Theorems & Definitions (63)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Conjecture 4
  • Lemma 5: HP33
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • Lemma 8
  • ...and 53 more