The Maximum of the Volume of a Cevian Simplex and its Parts
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910121513189376 |
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| author | Aliyev, Yagub N. |
| author_facet | Aliyev, Yagub N. |
| contents | The cevian triangle corresponding to an interior point $M$ of a triangle is the triangle determined by the feet of the three cevians concurrent at $M$. It is known that the area of the cevian triangle for an interior point $M$ of a triangle is at most $\frac{1}{4}$ of the area of the triangle, with maximum attained when $M$ is the triangle's centroid. This can be generalized from triangles to $n$-dimensional simplices, with $\frac{1}{4}$ replaced by $\frac{1}{n^n}$, using barycentric coordinates. We also use this method to solve two optimization problems about the parts of this simplex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Maximum of the Volume of a Cevian Simplex and its Parts Aliyev, Yagub N. Metric Geometry 51M04, 51M05, 51M16, 51M25 The cevian triangle corresponding to an interior point $M$ of a triangle is the triangle determined by the feet of the three cevians concurrent at $M$. It is known that the area of the cevian triangle for an interior point $M$ of a triangle is at most $\frac{1}{4}$ of the area of the triangle, with maximum attained when $M$ is the triangle's centroid. This can be generalized from triangles to $n$-dimensional simplices, with $\frac{1}{4}$ replaced by $\frac{1}{n^n}$, using barycentric coordinates. We also use this method to solve two optimization problems about the parts of this simplex. |
| title | The Maximum of the Volume of a Cevian Simplex and its Parts |
| topic | Metric Geometry 51M04, 51M05, 51M16, 51M25 |
| url | https://arxiv.org/abs/2507.10067 |