On the maximum product of distances of diameter $2$ point sets
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866908871578091520 |
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| author | Cambie, Stijn Decadt, Arne Dong, Yanni Hu, Tao Tang, Quanyu |
| author_facet | Cambie, Stijn Decadt, Arne Dong, Yanni Hu, Tao Tang, Quanyu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07088 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the maximum product of distances of diameter $2$ point sets Cambie, Stijn Decadt, Arne Dong, Yanni Hu, Tao Tang, Quanyu Combinatorics Complex Variables Metric Geometry 52C35, 30C15, 52A40, 52A10, 05C35, 05B30 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. |
| title | On the maximum product of distances of diameter $2$ point sets |
| topic | Combinatorics Complex Variables Metric Geometry 52C35, 30C15, 52A40, 52A10, 05C35, 05B30 |
| url | https://arxiv.org/abs/2603.07088 |