Strictly convex Wulff shapes and 𝐶¹ convex integrands

Han, Huhe; Nishimura, Takashi

doi:10.1090/proc/13510

Strictly convex Wulff shapes and convex integrands

Authors: Huhe Han and Takashi Nishimura
Journal: Proc. Amer. Math. Soc. 145 (2017), 3997-4008
MSC (2010): Primary 52A20, 52A55, 82D25
DOI: https://doi.org/10.1090/proc/13510
Published electronically: April 7, 2017
MathSciNet review: 3665051
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class . Moreover, applications of this result are given.

[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, https://doi.org/10.4171/RMI/559
[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 𝑅⁴ is Φ-minimizing, Math. Ann. 289 (1991), no. 2, 341–354. MR 1092180, https://doi.org/10.1007/BF01446576
[8] Takashi Nishimura, Normal forms for singularities of pedal curves produced by non-singular dual curve germs in 𝑆ⁿ, Geom. Dedicata 133 (2008), 59–66. MR 2390068, https://doi.org/10.1007/s10711-008-9233-5
[9] Takashi Nishimura, Singularities of pedal curves produced by singular dual curve germs in 𝑆ⁿ, 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, https://doi.org/10.5427/jsing.2010.2j
[11] Takashi Nishimura and Yu Sakemi, View from inside, Hokkaido Math. J. 40 (2011), no. 3, 361–373. MR 2883496, https://doi.org/10.14492/hokmj/1319595861
[12] Takashi Nishimura and Yu Sakemi, Topological aspect of Wulff shapes, J. Math. Soc. Japan 66 (2014), no. 1, 89–109. MR 3161393, https://doi.org/10.2969/jmsj/06610089
[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, https://doi.org/10.1112/plms/s3-50.2.370
[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 493671, https://doi.org/10.1090/S0002-9904-1978-14499-1
[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