Sets of tetrahedra, defined by maxima of distance functions

Rouyer, Joël; Vîlcu, Costin

doi:10.2478/v10309-012-0049-9

Abstract

We study tetrahedra and the space of tetrahedra from the viewpoint of local and global maxima for intrinsic distance functions

References

[1] P. K. Agarwal, B. Aronov, J. O’Rourke and C. A. Schevon, Star unfolding of a polytope with applications, SIAM J. Comput. 26 (1997), 1689-171310.1137/S0097539793253371
Search in Google Scholar
[2] A. D. Alexandrov, Die innere Geometrie cler konvexen Flächen, Akademie-Verlag, Berlin, 1955
Search in Google Scholar
[3] A. D. Alexandrov, Convex Polyhedra, Springer-Verlag, Berlin, 2005
Search in Google Scholar
[4] B. Aronov and J. O’Rourke, Non overlap of the star unfolding, Discrete Comput. Geom. 8 (1992), 219-25010.1007/BF02293047
Search in Google Scholar
[5] H. Buseman, Convetf surfaces, Dover, New York, 2008, originally published in 1958 by Interscience Publishers, Inc.
Search in Google Scholar
[6] H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, Springer-Verlag, New York, 199110.1007/978-1-4612-0963-8
Search in Google Scholar
[7] E. D. Demaine and J. O’Rourke, Geometric folding algorithms. Lineages, Origami, Polyhedra, Cambridge University Press, 200710.1017/CBO9780511735172
Search in Google Scholar
[8] J. Itoh, C. Nara and C. Vilcu, Continuous foldings of convex polyhedra, to appear
Search in Google Scholar
[9] J. Itoh and C. Vilcu, Farthest points and cut loci on some degenerate convex surfaces, J. Geom. 80 (2004), 106-12010.1007/s00022-004-1652-3
Search in Google Scholar
[10] J. M. Lee, Manifold and differentiable geometry, Graduate Studies in Mathematics 107., Providence, 2004
Search in Google Scholar
[11] J Rouyer, Antipodes sur le tétraèdre regulier, J. Geom. 77 (2003), 152-17010.1007/s00022-003-1617-y
Search in Google Scholar
[12] J. Rouyer, On antipodes on a convex polyhedron, Adv. Geom. 5 (2005), 497-50710.1515/advg.2005.5.4.497
Search in Google Scholar
[13] J. Rouyer, On antipodes on a convex polyhedron (II), Adv. Geom. 10 (2010), 403-41710.1515/advgeom.2010.014
Search in Google Scholar
[14] J. Rouyer and T. Sari, As many antipodes as vertices on convex polyhedra, Adv. Geom. 12 (2012), 43-6110.1515/advgeom.2011.030
Search in Google Scholar
[15] T. Sakai, Riemannian Geometry, Translation of Mathematical Mono- graphs 149, Amer. Math. Soc. 199610.1090/mmono/149
Search in Google Scholar
[16] M. Sharir and A. Schorr, On shortest paths in polyhedral spaces, SIAM J. Comput. 15 (1986), 193-21510.1137/0215014
Search in Google Scholar
[17] C. Vilcu, Properties of the farthest point mapping on convex surfaces, Rev. Roum. Math. Pures Appi. 51 (2006), 125-134
Search in Google Scholar
[18] C. Vilcu and T. Zamfirescu, Multiple farthest points on Alexandrov surfaces, Adv. Geom. 7 (2007), 83-10010.1515/ADVGEOM.2007.006
Search in Google Scholar
[19] V. A. Zalgaller, An isoperimetric problem for tetrahedra, J. Math. Sci. 140 (2007), 511-52710.1007/s10958-007-0431-8
Search in Google Scholar
[20] T. Zamfirescu, Farthest points on convex surfaces, Math. Z. 226 (1997), 623-63010.1007/PL00004360
Search in Google Scholar
[21] Tudor Zamfirescu, Extreme points of the distance function on a convex surface, Trans. Amer. Math. Soc. 350 (1998), 1395-1406. 10.1090/S0002-9947-98-02106-0
Search in Google Scholar

Sets of tetrahedra, defined by maxima of distance functions

Abstract

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