Journal of the Mathematical Society of Japan

Self-dual Wulff shapes and spherical convex bodies of constant width ${\pi}/{2}$

Huhe HAN and Takashi NISHIMURA

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For any Wulff shape, its dual Wulff shape is naturally defined. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, it is shown that a Wulff shape is self-dual if and only if the spherical convex body induced by it is of constant width ${\pi}/{2}$.

Article information

J. Math. Soc. Japan, Volume 69, Number 4 (2017), 1475-1484.

First available in Project Euclid: 25 October 2017

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Zentralblatt MATH identifier

Primary: 52A55: Spherical and hyperbolic convexity

Wulff shape dual Wulff shape self-dual Wulff shape spherical convex body width constant width Lune thickness diameter spherical polar set


HAN, Huhe; NISHIMURA, Takashi. Self-dual Wulff shapes and spherical convex bodies of constant width ${\pi}/{2}$. J. Math. Soc. Japan 69 (2017), no. 4, 1475--1484. doi:10.2969/jmsj/06941475.

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  • H. Han and T. Nishimura, Strictly convex Wulff shapes and $C^1$ convex integrands, Proc. Amer. Math. Soc., 145 (2017), 3997–4008.
  • M. Lassak, Width of spherical convex bodies, Aequationes Math., 89 (2015), 555–567.
  • M. Lassak, Reduced spherical polygons, Colloq. Math., 138 (2015), 205–216.
  • H. Maehara, Geometry of Circles and Spheres, Asakura Publishing, 1998 (in Japanese).
  • T. Nishimura and Y. Sakemi, Topological aspect of Wulff shapes, J. Math. Soc. Japan, 66 (2014), 89–109.
  • J. E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568–588.
  • G. Wulff, Zur frage der geschwindindigkeit des wachstrums und der auflösung der krystallflachen, Z. Kristallographine und Mineralogie, 34 (1901), 449–530.