On Axial Symmetry in Convex Bodies
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911541788409856 |
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| author | Goenka, Ritesh Moore, Kenneth Sun, Wen Rui White, Ethan Patrick |
| author_facet | Goenka, Ritesh Moore, Kenneth Sun, Wen Rui White, Ethan Patrick |
| contents | For a two-dimensional convex body, the Kovner-Besicovitch measure of symmetry is defined as the volume ratio of the largest centrally symmetric body contained inside the body to the original body. A classical result states that the Kovner-Besicovitch measure is at least $2/3$ for every convex body and equals $2/3$ for triangles. Lassak showed that an alternative measure of symmetry, i.e., symmetry about a line (axiality) has a value of at least $2/3$ for every convex body. However, the smallest known value of the axiality of a convex body is around $0.81584$, achieved by a convex quadrilateral. We show that every plane convex body has axiality at least $\frac{2}{41}(10 + 3 \sqrt{2}) \approx 0.69476$, thereby establishing a separation with the central symmetry measure. Moreover, we find a family of convex quadrilaterals with axiality approaching $\frac{1}{3}(\sqrt{2}+1) \approx 0.80474$. We also establish improved bounds for a ``folding" measure of axial symmetry for plane convex bodies. Finally, we establish improved bounds for a generalization of axiality to high-dimensional convex bodies. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_12597 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Axial Symmetry in Convex Bodies Goenka, Ritesh Moore, Kenneth Sun, Wen Rui White, Ethan Patrick Metric Geometry Computational Geometry 52A10, 52A38 (Primary) 52A20, 52A41 (Secondary) For a two-dimensional convex body, the Kovner-Besicovitch measure of symmetry is defined as the volume ratio of the largest centrally symmetric body contained inside the body to the original body. A classical result states that the Kovner-Besicovitch measure is at least $2/3$ for every convex body and equals $2/3$ for triangles. Lassak showed that an alternative measure of symmetry, i.e., symmetry about a line (axiality) has a value of at least $2/3$ for every convex body. However, the smallest known value of the axiality of a convex body is around $0.81584$, achieved by a convex quadrilateral. We show that every plane convex body has axiality at least $\frac{2}{41}(10 + 3 \sqrt{2}) \approx 0.69476$, thereby establishing a separation with the central symmetry measure. Moreover, we find a family of convex quadrilaterals with axiality approaching $\frac{1}{3}(\sqrt{2}+1) \approx 0.80474$. We also establish improved bounds for a ``folding" measure of axial symmetry for plane convex bodies. Finally, we establish improved bounds for a generalization of axiality to high-dimensional convex bodies. |
| title | On Axial Symmetry in Convex Bodies |
| topic | Metric Geometry Computational Geometry 52A10, 52A38 (Primary) 52A20, 52A41 (Secondary) |
| url | https://arxiv.org/abs/2309.12597 |