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Strictly convex Wulff shapes and $ C^1$ convex integrands

Authors: Huhe Han and Takashi Nishimura
Journal: Proc. Amer. Math. Soc. 145 (2017), 3997-4008
MSC (2010): Primary 52A20, 52A55, 82D25
Published electronically: April 7, 2017
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Abstract: In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class $ C^1$. Moreover, applications of this result are given.

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  • [1] Michael F. Barnsley, Fractals everywhere, 2nd ed., Academic Press Professional, Boston, MA, 1993. Revised with the assistance of and with a foreword by Hawley Rising, III. MR 1231795
  • [2] Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797
  • [3] Yoshikazu Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. A level set approach. MR 2238463
  • [4] H. Han and T. Nishimura, The spherical dual transform is an isometry for spherical Wulff shapes, to appear in Studia Math.
  • [5] Shyuichi Izumiya and Farid Tari, Projections of hypersurfaces in the hyperbolic space to hyperhorospheres and hyperplanes, Rev. Mat. Iberoam. 24 (2008), no. 3, 895-920. MR 2490202,
  • [6] Daisuke Kagatsume and Takashi Nishimura, Aperture of plane curves, J. Singul. 12 (2015), 80-91. MR 3317141
  • [7] Frank Morgan, The cone over the Clifford torus in $ {\bf R}^4$ is $ \Phi $-minimizing, Math. Ann. 289 (1991), no. 2, 341-354. MR 1092180,
  • [8] Takashi Nishimura, Normal forms for singularities of pedal curves produced by non-singular dual curve germs in $ S^n$, Geom. Dedicata 133 (2008), 59-66. MR 2390068,
  • [9] Takashi Nishimura, Singularities of pedal curves produced by singular dual curve germs in $ S^n$, Demonstratio Math. 43 (2010), no. 2, 447-459. MR 2668487
  • [10] T. Nishimura, Singularities of one-parameter pedal unfoldings of spherical pedal curves, J. Singul. 2 (2010), 160-169. MR 2763024,
  • [11] Takashi Nishimura and Yu Sakemi, View from inside, Hokkaido Math. J. 40 (2011), no. 3, 361-373. MR 2883496,
  • [12] Takashi Nishimura and Yu Sakemi, Topological aspect of Wulff shapes, J. Math. Soc. Japan 66 (2014), no. 1, 89-109. MR 3161393,
  • [13] A. Pimpinelli and J. Villain, Physics of Crystal Growth, Monographs and Texts in Statistical Physics, Cambridge University Press, Cambridge New York, 1998.
  • [14] S. A. Robertson and M. C. Romero Fuster, The convex hull of a hypersurface, Proc. London Math. Soc. (3) 50 (1985), no. 2, 370-384. MR 772718,
  • [15] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521
  • [16] Jean E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), no. 4, 568-588. MR 0493671
  • [17] J. E. Taylor, J. W. Cahn, and C. A. Handwerker, Geometric models of crystal growth, Acta Metallurgica et Materialia, 40(1992), 1443-1474.
  • [18] G. Wulff, Zur frage der geschwindindigkeit des wachstrums und der auflösung der krystallflachen, Z. Kristallographine und Mineralogie, 34(1901), 449-530.
  • [19] V. M. Zakaljukin, Singularities of convex hulls of smooth manifolds, Funkcional. Anal. i Priložen. 11 (1977), no. 3, 76-77 (Russian). MR 0458475

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Additional Information

Huhe Han
Affiliation: Graduate School of Environment and Information Sciences,Yokohama National University, Yokohama 240-8501, Japan

Takashi Nishimura
Affiliation: Research Institute of Environment and Information Sciences, Yokohama National University, Yokohama 240-8501, Japan

Keywords: Wulff shape, convex integrand, convex body, dual Wulff shape, spherical Wulff shape, spherical convex body, spherical dual Wulff shape.
Received by editor(s): January 24, 2016
Received by editor(s) in revised form: September 22, 2016, and September 26, 2016
Published electronically: April 7, 2017
Communicated by: Ken Ono
Article copyright: © Copyright 2017 American Mathematical Society

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