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Approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces
Metric Geometry · Submitted 26 Oct 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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. B\"or\"oczky ({\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 $1\leq k\leq d-2$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. | contribution 0.80 | 0.96 | 0.47 | asserted 1.00 |
| s1 | The metrics considered include the intrinsic volume difference and the Hausdorff metric. | method 0.76 | 0.52 | 0.04 | asserted 0.98 |
| s2 | 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. | result 0.67 | 0.91 | 0.60 | asserted 1.00 |
| s3 | For arbitrarily positioned polytopes, we also improve a special case of a result of K. | result 0.83 | 0.91 | 0.13 | asserted 0.99 |
| s4 | J. B\"or\"oczky ({\it J. Approx. | background 0.68 | 0.07 | 0.02 | unclear 0.97 |
| s5 | Theory}, 2000) by a factor of dimension. | result 0.81 | 0.63 | 0.08 | asserted 0.75 |
| s6 | This paper addresses a question of P. M. | background 0.81 | 0.17 | 0.02 | asserted 0.71 |
| s7 | 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 $1\leq k\leq d-2$. | background 0.74 | 0.09 | 0.04 | asserted 0.78 |
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The paper
Original abstract
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. B\"or\"oczky ({\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 $1\leq k\leq d-2$.
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How the literature uses it
1 arXiv paper cite this in 19 sentences
Papers citing it · 12 sentences without a confident label
Uses its method or tool 1 paper
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The combinatorial inequalities [equation] and [equation] which are due to Hinman [ref] , allowed the authors in [ref] to pass from estimates involving vertices or facets to estimates involving $k$-faces.
Dual volume approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces · Introduction · 0.86
Compares with it 1 paper
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This is why the minimum must appear in Theorem [ref] but not in the results of [ref] .
Dual volume approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces · General nonasymptotic lower bounds · 0.97
Cites it as background 1 paper
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For a convex body $K⊂^d$ and $N∈ℕ$, we set [equation] and [equation] For background on the results on the approximation of convex bodies by polytopes which are most closely related to this paper, we refer the reader to, e.g., [ref] and the references therein.
Dual volume approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces · Background and notation · 1.00
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