Approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915579356512256 |
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| author | Hoehner, Steven Schütt, Carsten Werner, Elisabeth |
| author_facet | Hoehner, Steven Schütt, Carsten Werner, Elisabeth |
| contents | We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and the Hausdorff metric. In the case of inscribed and circumscribed polytopes, our main results extend the previously obtained bounds from $k=0$ and $k=d-1$, respectively, to half of the $f$-vector of the approximating polytope. For arbitrarily positioned polytopes, we also improve a special case of a result of K. J. Böröczky ({\it J. Approx. Theory}, 2000) by a factor of dimension. This paper addresses a question of P. M. Gruber ({\it Convex and Discrete Geometry}, p. 216), who asked for results on the approximation of convex bodies by polytopes with a fixed number of $k$-faces when $1\leq k\leq d-2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22771 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces Hoehner, Steven Schütt, Carsten Werner, Elisabeth Metric Geometry 52A27 (Primary) 52A39, 52B11 (Secondary) We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and the Hausdorff metric. In the case of inscribed and circumscribed polytopes, our main results extend the previously obtained bounds from $k=0$ and $k=d-1$, respectively, to half of the $f$-vector of the approximating polytope. For arbitrarily positioned polytopes, we also improve a special case of a result of K. J. Böröczky ({\it J. Approx. Theory}, 2000) by a factor of dimension. This paper addresses a question of P. M. Gruber ({\it Convex and Discrete Geometry}, p. 216), who asked for results on the approximation of convex bodies by polytopes with a fixed number of $k$-faces when $1\leq k\leq d-2$. |
| title | Approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces |
| topic | Metric Geometry 52A27 (Primary) 52A39, 52B11 (Secondary) |
| url | https://arxiv.org/abs/2510.22771 |