Abstract

Fixed a set of finitely many segments in Rd\mathbb{R}^d we look for a connected system of these segments whose convex hull has the greatest d-dimensional volume, solving the problem when d and d+1 segments are considered. Some remarks about the analogous maximization problem on the set of the continuous and rectifiable curves with fixed length are also proposed.

Author information

Contact details are reproduced from the original publication and may be historical.

Michele Gori

Dip. di Matematica "L. Tonelli", Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

michele.gori@unifi.it

Keywords

  • Convex hull
  • system of segments

Mathematics Subject Classification

52A40

References

  1. L. Ambrosio, P. Tilli: Topics on Analysis in Metric Spaces, Oxford Lecture Series in Mathematics and its Applications 25, Oxford University Press, Oxford (2004).https://doi.org/10.1093/oso/9780198529385.001.0001Google Scholar
  2. A. Bezdek, K. Bezdek: On a discrete Dido-type question, Elem. Math. 4 (1989) 92–100.Google Scholar
  3. K. Bezdek, P. Brass, H. Harborth: Maximum convex hull of connected systems of segments and polyominoes, Contr. Alg. Geom. 35 (1994) 37–43.Google Scholar
  4. H. T. Croft, K. J. Falconer, R. K. Guy: Unsolved Problems in Geometry, Springer, New York (1991).https://doi.org/10.1007/978-1-4612-0963-8Google Scholar
  5. E. Egerváry: On the smallest convex cover of a simple arc of a space curve, Publ. Math., Debrecen 1 (1949) 65–70.https://doi.org/10.5486/PMD.1950.1.2-4.01Google Scholar
  6. M. G. Krein, A. A. Nudel’man: The Markov Moment Problem and Extremal Problems, Transl. Math. Monographs 50, Amer. Math. Soc, Providence (1977).https://doi.org/10.1090/mmono/050Google Scholar
  7. A. A. Nudel’man: Isoperimetric problems for the convex hulls of poligonal lines and curves in multidimensional spaces, Math. USSR, Sb. 25 (1975) 276–294.https://doi.org/10.1070/SM1975v025n02ABEH002209Google Scholar
  8. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der mathematischen Wissenschaften 317, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  9. G. C. Shephard: Combinatorial properties of associated zonotopes, Can. J. Math. 26 (1974) 302–321.https://doi.org/10.4153/CJM-1974-032-5Google Scholar
  10. I. J. Shoenberg: An isoperimetric inequality for closed curves convex in even dimensional Euclidean spaces, Acta Math. 91 (1954) 265–274.https://doi.org/10.1007/BF02393429Google Scholar
  11. A. Siegel: Some Dido-type inequalities, Elem. Math. 56 (2001) 17–20.https://doi.org/10.1007/s000170050085Google Scholar
  12. V. A. Zalgaller: Extremal problems concerning the convex hull of a space curve, St. Petersbg. Math. J. 8 (1997) 369–379.Google Scholar

Suggested citation

M. Gori. “On a Maximization Problem for the Convex Hull of Connected Systems of Segments.” Journal of Convex Analysis 14 (2007), No. 1, 49–68. https://doi.org/10.68381/jca14005

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.