Convex Domains of Given Diameter with Greatest Volume

  • Uwe Klemt
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 124)


In this paper we study the following geometric optimization problem: What are the convex domains of given diameter with greatest volume in the n-dimensional Euclidean space (n ≥ 2)? By using the plane of support of a convex domain we formulate the above mentioned question analytically. This leads to a multidimensional variational problem in parametric form with respect to a class of state functions and associated control vectors of Grassmann coordinates fulfilling certain state restrictions, control restrictions and boundary conditions. In order to prove the conjecture that circle and ball, respectively, are domains solving the problem we apply a generalized duality theory in the sense of R. Klötzler. On the basis of this theory first-order necessary conditions and second-order sufficiency conditions for an auxiliary finite-dimensional parametric optimization problem are verified.


Convex Domain Maple Versus Weak Local Minimizer Strong Local Minimizer Control Constraint State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bertini, E.: Einführung in die projektive Geometrie mehrdimensionaler Räume. L. W. Seidel & Sohn, Wien, 1924.Google Scholar
  2. [2]
    Blaschke, W.: Kreis und Kugel. Teubner-Verlag, Leipzig, 1916.zbMATHGoogle Scholar
  3. [3]
    Bonnesen, T.; Fenchel, W.: Theorie der konvexen Körper. Springer-Verlag, Berlin-Heidelberg-New York, 1974.zbMATHCrossRefGoogle Scholar
  4. [4]
    Fiacco, A. V.; McCormic, G. P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM, Philadelphia, 1990.zbMATHCrossRefGoogle Scholar
  5. [5]
    Fuchs, G.: Lagrangesche Variationsprobleme in Parameterdarstellung. PhD thesis, Universität Halle-Wittenberg, 1969.Google Scholar
  6. [6]
    Jaglom, J. M.; Boltyanski, W. G.: Konvexe Figuren. Deutscher Verlag der Wissenschaften, Berlin, 1956.Google Scholar
  7. [7]
    Klemt, U.: A Remark on the Paper “Convex Domains of Given Thickness with Smallest Surface Area” by R. Klötzler. Reihe Mathematik M-08/1995, Technische Universität Cottbus, 1995.Google Scholar
  8. [8]
    Klötzler, R.: Models and Applications of Duality in Optimal Control. In Proc. IFIP Working Conference on Recent Advances in System Modeling and Optimization, Hanoi, 1983.Google Scholar
  9. [9]
    Klötzler, R.: Konvexe Bereiche kleinster Oberfläche bei gegebener Dicke. Z. Anal. Anw., 4(4):373–383, 1984.Google Scholar
  10. [10]
    Kubota, T.: Über konvex-geschlossene Mannigfaltigkeiten im n-dimensionalen Raume. Sci. Rep. Tôhoku Univ., 14:85–99, 1925.zbMATHGoogle Scholar
  11. [11]
    Madelung, E.: Die mathematischen Hilfsmittel des Physikers. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953.zbMATHGoogle Scholar
  12. [12]
    Minkowski. H.: Theorie der konvexen Körper. In Gesammelte Abhandlungen, pages 131–229. Teubner-Verlag, Leipzig, 1911.Google Scholar

Copyright information

© Springer Basel AG 1998

Authors and Affiliations

  • Uwe Klemt
    • 1
  1. 1.Lehrstuhl OptimierungBTU CottbusCottbusGermany

Personalised recommendations