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Volume of Convex Hull of Two Bodies and Related Problems

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Discrete Geometry and Symmetry (GSC 2015)

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Abstract

In this paper we deal with problems concerning the volume of the convex hull of two “connecting” bodies. After a historical background we collect some results, methods and open problems, respectively.

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References

  1. L. Lindelöf, Propriétés générales des polyèdres qui, sous une étendue superficielle donnée, renferment le plus grand volume. Bull. Acad. Sci. St. Petersburg 14, 258–269 (1869). Math. Ann. 2, 150–159 (1870)

    Google Scholar 

  2. P. Gruber, Convex and Discrete Geometry, Grundlehren der mathematischen Wissenschaften 336 (Springer, Berlin, Heidelberg, 2007)

    Google Scholar 

  3. K. Böröczky Jr., K. Böröczky, Isoperimetric problems for polytopes with a given number of vertices. Mathematika 43, 237–254 (1996)

    Article  MathSciNet  Google Scholar 

  4. H.T. Croft, K.J. Falconer, R.K. Guy, Unsolved Problems in Geometry, vol. 2 (Springer, New York, 1991)

    MATH  Google Scholar 

  5. Florian, A., in Handbook of Convex Geometry, ed. by P.M. Gruber, J.M. Wills. Extremum Problems for Convex Discs and Polyhedra (North-Holland Publishing Co., Amsterdam, 1993)

    Chapter  Google Scholar 

  6. P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry (Springer, New York, 2005)

    MATH  Google Scholar 

  7. B. Kind, P. Kleinschmidt, On the maximal volume of convex bodies with few vertices. J. Combin. Theory Ser. A 21, 124–128 (1976)

    Article  MathSciNet  Google Scholar 

  8. A. Klein, M. Wessler, The largest small \(n\)-dimensional polytope with \(n + 3\) vertice. J. Combin. Theory Ser. A 102, 401–409 (2003)

    Article  MathSciNet  Google Scholar 

  9. A. Klein, M. Wessler, A correction to “The largest small \(n\)-dimensional polytope with \(n + 3\) vertices” [J. Combin. Theory Ser. A 102, 401–409 (2003)]. J. Combin. Theory Ser. A 112, 173–174 (2005)

    Google Scholar 

  10. L. Fejes-Tóth, Regular Figures (The Macmillan Company, New York, 1964)

    MATH  Google Scholar 

  11. J.D. Berman, K. Hanes, Volumes of polyhedra inscribed in the unit sphere in \(E^3\). Math. Ann. 188, 78–84 (1970)

    Article  MathSciNet  Google Scholar 

  12. N. Mutoh, The polyhedra of maximal volume inscribed in the unit sphere and of minimal volume circumscribed about the unit sphere. JCDCG, Lect. Notes Comput. Sci. 2866, 204–214 (2002)

    Article  MathSciNet  Google Scholar 

  13. A.M. Macbeath, An extremal property of the hypersphere. Proc. Cambridge Philos. Soc. 47, 245–247 (1951)

    Article  MathSciNet  Google Scholar 

  14. C.A. Rogers, G.C. Shephard, Convex bodies associated with a given convex body. J. London Math. Soc. 33, 270–281 (1958)

    Article  MathSciNet  Google Scholar 

  15. C.A. Rogers, G.C. Shephard, Some extremal problems for convex bodies. Mathematika 5(2), 93–102 (1958)

    Article  MathSciNet  Google Scholar 

  16. Á.G. Horváth, Maximal convex hull of connecting simplices. Stud. Univ. Zilina 22/1, 7–19 (2008)

    Google Scholar 

  17. Á.G. Horváth, Z. Lángi, On the volume of the convex hull of two convex bodies. Monatshefte für Mathematik 174(2), 219–229 (2014)

    Article  MathSciNet  Google Scholar 

  18. Á.G. Horváth, On an extremal problem connected with simplices. Beiträge zur Geometrie und Algebra 55(2), 415–428 (2014)

    Article  MathSciNet  Google Scholar 

  19. R. Bowen, S. Fisk, Generation of triangulations of the sphere. Math. Comp. 21, 250–252 (1967)

    MathSciNet  MATH  Google Scholar 

  20. M. Goldberg, The isoperimetric problem for polyhedra. Tóhoku Math. J. 40, 226–236 (1935)

    MATH  Google Scholar 

  21. D.W. Grace, Search for largest polyhedra. Math. Comp. 17, 197–199 (1963)

    Article  Google Scholar 

  22. A. Florian, Integrale auf konvexen polyedern period. Math. Hungar. 1, 243–278 (1971)

    Article  MathSciNet  Google Scholar 

  23. Á.G. Horváth, On the icosahedron inequality of László Fejes-Tóth. J. Math. Inequalities 10/2, 521–539 (2016)

    Google Scholar 

  24. Á.G. Horváth, Z. Lángi, Maximum volume polytopes inscribed in the unit sphere. Monaschefte für Math. 181/2, 341–354 (2016)

    Article  MathSciNet  Google Scholar 

  25. Böröczky, K., On an extremum property of the regular simplex in \(\cal{S}^d\). Colloq. Math. Soc. János Bolyai48 Intuitive Geometry, pp. 117–121. Siófok (1985)

    Google Scholar 

  26. J.W. Milnor, in The geometry and Topology of 3-Manifolds, ed. by W.P. Thurston, Computation of volume (Lecture notes at Princeton University, 1977–1978)

    Google Scholar 

  27. J.W. Milnor, Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (N.S.) 6/1, 9–24 (1982)

    Article  MathSciNet  Google Scholar 

  28. U. Haagerup, H.J. Munkholm, Simplices of maximal volume in hyperbolic n-space. Acta. Math. 147, 1–12 (1981)

    Article  MathSciNet  Google Scholar 

  29. B. Grünbaum, Convex Polytopes, 2nd edn. (Springer, New York, 2003)

    Book  Google Scholar 

  30. G.M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics 152 (Springer, New York, 1995)

    Google Scholar 

  31. C.W. Lee, Regular triangulation of convex polytopes. Dimacs Ser. Discrete Math. Theor. Comput. Sci.4, 443–456 (1991)

    Google Scholar 

  32. V. Kaibel, A. Wassmer, Automorphism Groups of Cyclic Polytopes, Chapter 8 of F. Lutz, Triangulated Manifolds with Few Vertices, Algorithms and Combinatorics (Springer, New York, to appear)

    Google Scholar 

  33. I. Fáry, L.Der Rédei, zentralsymmetrische Kern und die zentralsymmetrische Hülle von konvexen Körpern. Math. Annalen. 122, 205–220 (1950)

    Article  MathSciNet  Google Scholar 

  34. H.-K. Ahn, P. Brass, C.-S. Shin, Maximum overlap and minimum convex hull of two convex polyhedra under translations. Comput. Geom. 40, 171–177 (2008)

    Article  MathSciNet  Google Scholar 

  35. H. Martini, Z. Mustafaev, Some applications of cross-section measures in Minkowski spaces. Period. Math. Hungar. 53, 185–197 (2006)

    Article  MathSciNet  Google Scholar 

  36. H. Martini, K. Swanepoel, Antinorms and Radon curves. Aequationes Math. 72, 110–138 (2006)

    Article  MathSciNet  Google Scholar 

  37. J. Alonso, C. Benítez, Orthogonality in normed linear spaces: a survey, II. Relat. Between Main Orthogonalities, Extracta Math. 4, 121–131 (1989)

    Google Scholar 

  38. K. Ball, Ellipsoids of maximal volume in convex bodies. Geom. Dedicata 41(2), 241–250 (1992)

    Article  MathSciNet  Google Scholar 

  39. R.J. Gardner, Geometric Tomography (Cambridge University Press, Cambridge, 1995)

    MATH  Google Scholar 

  40. P. Gronchi, M. Longinetti, Affine regular polygons as extremals of area functionals. Discrete Comput. Geom. 39, 273–297 (2008)

    Article  MathSciNet  Google Scholar 

  41. M. Henk, Löwner-John ellipsoids. Doc. Math. Extra volume ISMP (2012), 95–106

    Google Scholar 

  42. F. John, in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948. Extremum Problems with Inequalities as Subsidiary Conditions (Interscience Publishers, Inc., New York, 1948), pp. 187–204

    Google Scholar 

  43. A.C. Thompson, Minkowski Geometry, Encyclopedia of Mathematics and its Applications 63 (Cambridge University Press, Cambridge, 1996)

    Google Scholar 

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Authors and Affiliations

  1. Department of Geometry, Budapest University of Technology, Egry József u. 1, Budapest, 1111, Hungary

    Ákos G. Horváth

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  1. Ákos G. Horváth
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Correspondence to Ákos G. Horváth .

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Editors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, New Zealand

    Marston D. E. Conder

  2. Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada

    Antoine Deza

  3. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

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Horváth, Á.G. (2018). Volume of Convex Hull of Two Bodies and Related Problems. In: Conder, M., Deza, A., Weiss, A. (eds) Discrete Geometry and Symmetry. GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham. https://doi.org/10.1007/978-3-319-78434-2_11

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