Abstract
The intersection L of two different non-opposite hemispheres G and H of the d-dimensional unit sphere \(S^d\) is called a lune. By the thickness of L we mean the distance of the centers of the \((d-1)\)-dimensional hemispheres bounding L. For a hemisphere G supporting a convex body \(C \subset S^d\) we define \(\mathrm{width}_G(C)\) as the thickness of the narrowest lune or lunes of the form \(G \cap H\) containing C. If \(\mathrm{width}_G(C) =w\) for every hemisphere G supporting C, we say that C is a body of constant width w. We present properties of these bodies. In particular, we prove that the diameter of any spherical body C of constant width w on \(S^d\) is w, and that if \(w < \frac{\pi }{2}\), then C is strictly convex. Moreover, we check when spherical bodies of constant width and constant diameter coincide.
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Lassak, M., Musielak, M. Spherical bodies of constant width. Aequat. Math. 92, 627–640 (2018). https://doi.org/10.1007/s00010-018-0558-3
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DOI: https://doi.org/10.1007/s00010-018-0558-3
Keywords
- Spherical convex body
- Spherical geometry
- Hemisphere
- Lune
- Width
- Constant width
- Thickness
- Diameter
- Extreme point