Generalization of Marion's Theorem: Volumes of Central Polytopes Obtained by Trisecting the Edges of Simplices
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916074264461312 |
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| author | Kazakov, Yu. V. |
| author_facet | Kazakov, Yu. V. |
| contents | The classical Marion's theorem states that the area of the central hexagon obtained by dividing each side of a triangle into three equal parts and connecting the interior division points is exactly $1/10$ of the area of the original triangle. In this paper, the construction is extended to an arbitrary n-dimensional simplex: each edge is divided into three equal parts, and through each n-1 vertices and the two interior division points of the opposite edge, two hyperplanes are drawn. It is shown that the resulting inner polytope has a surprisingly simple volume formula with the volume is $1/C(n,2n+1)$ times the volume of the original simplex. The proof uses a probabilistic interpretation with exponential random variables and reduces the geometric problem to a combinatorial binomial sum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_02087 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalization of Marion's Theorem: Volumes of Central Polytopes Obtained by Trisecting the Edges of Simplices Kazakov, Yu. V. Metric Geometry 51M20 I.3.5 The classical Marion's theorem states that the area of the central hexagon obtained by dividing each side of a triangle into three equal parts and connecting the interior division points is exactly $1/10$ of the area of the original triangle. In this paper, the construction is extended to an arbitrary n-dimensional simplex: each edge is divided into three equal parts, and through each n-1 vertices and the two interior division points of the opposite edge, two hyperplanes are drawn. It is shown that the resulting inner polytope has a surprisingly simple volume formula with the volume is $1/C(n,2n+1)$ times the volume of the original simplex. The proof uses a probabilistic interpretation with exponential random variables and reduces the geometric problem to a combinatorial binomial sum. |
| title | Generalization of Marion's Theorem: Volumes of Central Polytopes Obtained by Trisecting the Edges of Simplices |
| topic | Metric Geometry 51M20 I.3.5 |
| url | https://arxiv.org/abs/2606.02087 |