The Complex Illumination Problem
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913548307791872 |
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| author | Rotem, Liran Schejter, Alon Slomka, Boaz A. |
| author_facet | Rotem, Liran Schejter, Alon Slomka, Boaz A. |
| contents | We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in C^n, as well as its (non-comparable) fractional version. A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in C^n is equal to 2^(n+1)-1 and that the fractional illumination number of the polydisc in C^n is 2^n. In addition, we verify both conjectures for the classes of complex zonotopes and zonoids. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_12021 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Complex Illumination Problem Rotem, Liran Schejter, Alon Slomka, Boaz A. Metric Geometry 52C17 (Primary) 52A21, 52A37 (Secondary) We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in C^n, as well as its (non-comparable) fractional version. A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in C^n is equal to 2^(n+1)-1 and that the fractional illumination number of the polydisc in C^n is 2^n. In addition, we verify both conjectures for the classes of complex zonotopes and zonoids. |
| title | The Complex Illumination Problem |
| topic | Metric Geometry 52C17 (Primary) 52A21, 52A37 (Secondary) |
| url | https://arxiv.org/abs/2410.12021 |