The farthest point map on the 4-cube
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908169934995456 |
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| author | Yamagishi, Yoshikazu |
| author_facet | Yamagishi, Yoshikazu |
| contents | We study the farthest point mapping on (the boundary of) the 4-cube with respect to the intrinsic metric, and its dynamics as a multivalued mapping. It is a piecewise rational map. It is more complicated than the one on the 3-cube, but it is shown that the limit set of the farthest point map on the 4-cube is the union of the diagonals of eight (3-cube) facets, like the farthest point map on the 3-cube whose limit set is the union of the six (square) facets. This is in contrast to the doubly covered simplices and (the boundary of) the regular 4-simplex, where the limit set is a finite set. If the source point is in the interior of a facet, its limit set is also in the facet.
The farthest point mapping is closely related to the star unfolding and source unfolding. We give a loose definition of star unfolding of the surface of a 4-dimensional polytope.
We also study the intrinsic radius and diameter of the 4-cube. It is expected that the intrinsic radius/diameter ratio of an n-cube is monotonically decreasing in dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The farthest point map on the 4-cube Yamagishi, Yoshikazu Metric Geometry Dynamical Systems 52B11, 37F10 We study the farthest point mapping on (the boundary of) the 4-cube with respect to the intrinsic metric, and its dynamics as a multivalued mapping. It is a piecewise rational map. It is more complicated than the one on the 3-cube, but it is shown that the limit set of the farthest point map on the 4-cube is the union of the diagonals of eight (3-cube) facets, like the farthest point map on the 3-cube whose limit set is the union of the six (square) facets. This is in contrast to the doubly covered simplices and (the boundary of) the regular 4-simplex, where the limit set is a finite set. If the source point is in the interior of a facet, its limit set is also in the facet. The farthest point mapping is closely related to the star unfolding and source unfolding. We give a loose definition of star unfolding of the surface of a 4-dimensional polytope. We also study the intrinsic radius and diameter of the 4-cube. It is expected that the intrinsic radius/diameter ratio of an n-cube is monotonically decreasing in dimension. |
| title | The farthest point map on the 4-cube |
| topic | Metric Geometry Dynamical Systems 52B11, 37F10 |
| url | https://arxiv.org/abs/2412.16862 |