From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911765077426176 |
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| author | Brandenberg, René von Dichter, Katherina Merino, Bernardo González |
| author_facet | Brandenberg, René von Dichter, Katherina Merino, Bernardo González |
| contents | For a Minkowski centered convex compact set $K$ we define $α(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\mathrm{conv} (K\cup (-K))$ and give a complete description of the possible values of $α(K)$ in the planar case in dependence of the Minkowski asymmetry of $K$. As a side product, we show that, if the asymmetry of $K$ is greater than the golden ratio, the boundary of $K$ intersects the boundary of its negative $-K$ always in exactly 6 points. As an application, we derive bounds for the diameter-width-ratio for pseudo-complete and complete sets, again in dependence of the Minkowski asymmetry of the convex bodies, tightening those depending solely on the dimension given in a recent result of Richter [10]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11460 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets Brandenberg, René von Dichter, Katherina Merino, Bernardo González Metric Geometry For a Minkowski centered convex compact set $K$ we define $α(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\mathrm{conv} (K\cup (-K))$ and give a complete description of the possible values of $α(K)$ in the planar case in dependence of the Minkowski asymmetry of $K$. As a side product, we show that, if the asymmetry of $K$ is greater than the golden ratio, the boundary of $K$ intersects the boundary of its negative $-K$ always in exactly 6 points. As an application, we derive bounds for the diameter-width-ratio for pseudo-complete and complete sets, again in dependence of the Minkowski asymmetry of the convex bodies, tightening those depending solely on the dimension given in a recent result of Richter [10]. |
| title | From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2306.11460 |