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Farthest points on flat surfaces

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We consider the distance function from an arbitrary point p on a closed flat surface, and determine the set \(F_{p}\) of all farthest points (i.e., points at maximal distance) from p.

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  1. Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry. Springer, New York (1991)

    Book  Google Scholar 

  2. Zamfirescu, T.: On some questions about convex surfaces. Math. Nach. 172, 313–324 (1995)

    Article  MathSciNet  Google Scholar 

  3. Vîlcu, C.: Properties of the farthest point mapping on convex surfaces. Rev. Roum. Math. Pures Appl. 51, 125–134 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Rouyer, J.: Antipodes sur un tétraèdre régulier. J. Geom. 77, 152–170 (2003)

    Article  MathSciNet  Google Scholar 

  5. Nikonorov, Y.G., Nikonorova, Y.V.: The intrinsic diameter of the surface of a parallelepiped. Discrete Comput. Geom. 40, 504–527 (2008)

    Article  MathSciNet  Google Scholar 

  6. Itoh, J.-I., Rouyer, J., Vîlcu, C.: Antipodal convex hypersurfaces. Indag. Math. New Ser. 19, 411–426 (2008)

    Article  MathSciNet  Google Scholar 

  7. Itoh, J.-I., Vîlcu, C.: What do cylinders look like? J. Geom. 95, 41–48 (2009)

    Article  MathSciNet  Google Scholar 

  8. Vîlcu, C.: On two conjectures of Steinhaus. Geom. Dedicata 79, 267–275 (2000)

    Article  MathSciNet  Google Scholar 

  9. Vîlcu, C., Zamfirescu, T.: Symmetry and the farthest point mapping on convex surfaces. Adv. Geom. 6, 345–353 (2006)

    Article  MathSciNet  Google Scholar 

  10. Rouyer, J.: Steinhaus conditions for convex polyhedra. In: Adiprasito, K., et al. (eds.) Convexity and Discrete Geometry Including Graph Theory. Springer Proceedings in Mathematics and Statistics, vol. 148, pp. 77–84. Springer, Cham (2016)

    Chapter  Google Scholar 

  11. Vîlcu, C., Zamfirescu, T.: Multiple farthest points on Alexandrov surfaces. Adv. Geom. 7, 83–100 (2007)

    Article  MathSciNet  Google Scholar 

  12. Rouyer, J., Vîlcu, C.: Farthest points on most Alexandrov surfaces. Adv. Geom. arXiv:1412.1465 [math.MG] (to appear)

  13. Burago, Y., Gromov, M., Perel’man, G.: A. D. Alexandrov spaces with curvature bounded below. Russ. Math. Surv. 47, 1–58 (1992). (English, Russian original)

    Article  Google Scholar 

  14. Shiohama, K.: An Introduction to the Geometry of Alexandrov Spaces. Lecture Notes Series, Seoul National University, vol. 8 (1992)

  15. Shiohama, K., Tanaka, M.: Cut loci and distance spheres on Alexandrov surfaces, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992). In: Séminaires Congreè, vol. 1, pp. 531–559. Société Mathématique de France, Paris (1996)

  16. Rouyer, J., Vîlcu, C.: The connected components of the space of Alexandrov surfaces. In: Ibadula, D., Veys, W. (eds.) Experimental and Theoretical Methods in Algebra, Geometry and Topology. Springer Proceedings in Mathematics and Statistics, vol. 96, pp. 249–254. Springer, Cham (2014)

    Google Scholar 

  17. Zamfirescu, T.: Points joined by three shortest paths on convex surfaces. Proc. Am. Math. Soc. 123, 3513–3518 (1995)

    Article  MathSciNet  Google Scholar 

  18. Zamfirescu, T.: Extreme points of the distance function on convex surfaces. Trans. Am. Math. Soc. 350, 1395–1406 (1998)

    Article  MathSciNet  Google Scholar 

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Correspondence to Costin Vîlcu.

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Rouyer, J., Vîlcu, C. Farthest points on flat surfaces. J. Geom. 109, 44 (2018).

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