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Plane Lorentzian and Fuchsian Hedgehogs

Published online by Cambridge University Press:  20 November 2018

Yves Martinez-Maure
Yves Martinez-Maure*
Affiliation:
Institut Mathématique de Jussieu - Paris Rive Gauche, Bâtiment Sophie Germain, Case 7012, Paris Cedex 13, France e-mail: yves.martinez-maure@imj-prg.fr
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Abstract

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Parts of the Brunn–Minkowski theory can be extended to hedgehogs, which are envelopes of families of affine hyperplanes parametrized by their Gauss map. F. Fillastre introduced Fuchsian convex bodies, which are the closed convex sets of Lorentz–Minkowski space that are globally invariant under the action of a Fuchsian group. In this paper, we undertake a study of plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the Fuchsian analogues of classical geometrical inequalities (analogues that are reversed as compared to classical ones).

Keywords

52A4052A5553A0453B30Fuchsian and Lorentzian hedgehogsevolutedualityconvolutionreversed isoperimetric inequalityreversed Bonnesen inequality

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 3 , 01 September 2015 , pp. 561 - 574
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Eggleston, H. G., Convexity. Cambridge Tracts in Mathematics and Mathematical Physics, 47, Cambridge University Press, New York, 1958.Google Scholar
[2] Fillastre, F., Fuchsianconvex bodies: basics of Brunn-Minkowski theory. Geom. Funct. Anal. 23(2013), no. 1, 295–333. http://dx.doi.org/10.1007/s00039-012-0205-4 Google Scholar
[3] Geppert, H., tJber den Brunn-MinkowskischenSatz. Math. Z. 42(1937), no., 1, 238–254. http://dx.doi.org/10.1007/BF01160076 Google Scholar
[4] Gôrtler, H., ErzeugungstiitzbarerBereiche I. Deutsche Math. 2(1937), 454–456.Google Scholar
[5] Gôrtler, H., ErzeugungstiitzbarerBereiche II. Deutsche Math. 3(1937), 189–200.Google Scholar
[6] Langevin, R., Levitt, G., and Rosenberg, H., Hérissons et multihérissons (enveloppes paramétrées par leur application de Gauss). In: Singularities (Warsaw, 1985), Banach Center Publ, 20, PWN, Warsaw, 1988, pp. 245–253.Google Scholar
[7] Lopez, R., Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int. Electron. J. Geom 7(2014), no. 1, 44–107.Google Scholar
[8] Martinez-Maure, Y., De nouvelles inégalités géométriques pour les hérissons. Arch. Math. (Basel) 72(1999), no. 6, 444–453. http://dx.doi.org/10.1007/s000130050354 Google Scholar
[9] Martinez-Maure, Y., A fractal protective hedgehog. DemonstratioMath. 34(2001), no. 1, 59–63.Google Scholar
[10] Martinez-Maure, Y., Geometric study of Minkowski differences of plane convex bodies. Canad. J. Math. 58(2006), no. 3, 600–624. http://dx.doi.org/10.41 53/CJM-2OO6-O2 5-X Google Scholar
[11] McMullen, P., Thepolytope algebra. Adv. Math. 78(1989), no. 1, 76–130. http://dx.doi.Org/10.101 6/0001-8708(89)90029-7 Google Scholar
[12] Osserman, R., Bonnesen-style isoperimetric inequalities. Am. Math. Monthly 86(1979), no. 1,1–29. http://dx.doi.org/10.2307/2320297 Google Scholar
[13] Schneider, R., Convex bodies: the Brunn-Minkowski theory. Second expanded éd.,Encyclopedia of Mathematics and its Applications, 151, Cambridge University Press, Cambridge, 2014.Google Scholar