On a geometric extremum problem for convex cones
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918168477302784 |
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| author | Mushkarov, Oleg Nikolov, Nikolai |
| author_facet | Mushkarov, Oleg Nikolov, Nikolai |
| contents | We discuss the optimization problem for minimizing the $(n-1)$-volume of the intersection of a convex cone $K$ in $\Bbb R^n$ with a hyperplane through a given point, first considered in \cite{We}. We give a geometric characterization of the stationary hyperplanes for this problem when $K$ is a hyperangle which partially answers a question posed in \cite{We}. Moreover, we study the location of the set $S$ of points for which there is a stationary hyperplane as well as the infimum of the $(n-1)$-volumes of cone segments of $K$ cut off by hyperplanes through a given boundary point of $K$. As a model example we study in detail the non-negative orthant of $\Bbb R^n$. In this case $S$ is its interior and we show that every point of $S$ lies in a unique stationary hyperplane, which we describe in terms of the unique real root of an irrational equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a geometric extremum problem for convex cones Mushkarov, Oleg Nikolov, Nikolai Metric Geometry 49K21, 51M16, 52A38 We discuss the optimization problem for minimizing the $(n-1)$-volume of the intersection of a convex cone $K$ in $\Bbb R^n$ with a hyperplane through a given point, first considered in \cite{We}. We give a geometric characterization of the stationary hyperplanes for this problem when $K$ is a hyperangle which partially answers a question posed in \cite{We}. Moreover, we study the location of the set $S$ of points for which there is a stationary hyperplane as well as the infimum of the $(n-1)$-volumes of cone segments of $K$ cut off by hyperplanes through a given boundary point of $K$. As a model example we study in detail the non-negative orthant of $\Bbb R^n$. In this case $S$ is its interior and we show that every point of $S$ lies in a unique stationary hyperplane, which we describe in terms of the unique real root of an irrational equation. |
| title | On a geometric extremum problem for convex cones |
| topic | Metric Geometry 49K21, 51M16, 52A38 |
| url | https://arxiv.org/abs/2502.19381 |