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Minimal ellipsoids and their duals

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Abstract

It is proved that “most” convex bodies in IEd touch the boundaries of their minimal circumscribed and their maximal inscribed ellipsoids in preciselyd(d+3)/2 points. A version of the former result shows that for “most” compact sets in IEd the corresponding optimal designs, i.e. probability measures with a certain extremal property, are concentrated ond(d+1)/2 points.

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References

  1. Bachem A., Groetschel M.,New aspects of polyhedral theory, In: B. Korte, ed.: Modern applied mathematics (optimization and operations research), Amsterdam, North-Holland 1982, 51–106.

    Google Scholar 

  2. Behrend F.,Über die kleinste umbeschriebene und die größte einbeschriebene Ellipse eines konvexen Bereiches, Math. Zeitschr.115 (1938), 374–411.

    MathSciNet  Google Scholar 

  3. Berger M.,Géométrie 3/convexes et polytopes, polyèdres réguliers aires et volumes, Paris, Cedic/Nathan 1978.

    MATH  Google Scholar 

  4. Brauner H.,Geometrie projektiver Räume I, Mannheim, Bibl. Inst. 1976.

  5. Chernoŭsko L.,Guaranteed ellipsoidal estimate of uncertainties in control problems, Wiss. Z. Techn. Hochsch. Leipzing4 (1980), 325–329.

    Google Scholar 

  6. Danzer L., Grünbaum B., Klee V.,Helly's theorem and its relatives, In: V.L. Klee, ed.: Convexity, Proc. Sympos. Pure Math. VII (Seattle 1961) 101–180, Providence R.I., Amer. Math. Soc. 1963.

    Google Scholar 

  7. Danzer L., Laugwitz D., Lenz H.,Über das Löwnersche Ellipsoid und sein Analogon unter den einem Eikörper einbeschriebenen Ellipsoiden, Arch. Math.8 (1975), 214–219

    Article  MathSciNet  Google Scholar 

  8. Firey W.J.,Some applications of means of convex bodies, Pacific J. Math.14 (1964), 53–60.

    MATH  MathSciNet  Google Scholar 

  9. Gel'fand I.M.,Lectures on linear algebra, New York, Interscience 1961.

    MATH  Google Scholar 

  10. Gruber P.M.,Stability of isometries, Trans. Amer. Math. Soc.,245 (1978), 263–277.

    Article  MATH  MathSciNet  Google Scholar 

  11. Gruber P.M.,Results of Baire category type in convexity, In J.E. Goodman, E. Lutwak, J. Malkevitch, R. Pollack, eds.: Discrete geometry and convexity, Ann. New York Acad. Sci.,440 (1985), 163–170.

  12. Gruber P.M., Höbinger J.,Kennzeichnungen von Ellipsoiden und Anwendungen, In: D. Laugwitz, ed.: Jahrb. Überbl. Math.,1976 (1976), 9–27, Mannheim, Bibl. Inst. 1976.

  13. Holmes R.B.,Geometrical functional analysis and its applications, New York, Springer 1975.

    Google Scholar 

  14. John F.,Extremum with inequalities as subsidiary conditions, In: K.O. Friedrichs, O. Neugebauer, J.J. Stoker, eds., Studies and essays, Courant's anniversary volume, New York, J. Wiley 1948.

    Google Scholar 

  15. Kadets M.I.,The geometry of normed spaces, Itogi Nauk Tekhn., Mat. Analiz.,13 (1975), 99–128—J. Soviet Math.,7 (1977), 953–973.

    Google Scholar 

  16. Karlin S., Studden W.J.,Tchebycheff systems: with applications in analysis and statistics, New York, Interscience 1966.

    MATH  Google Scholar 

  17. Khachyan L.G.,A polynomial algorithm in linear programming, Dokl. Akad. Nauk SSSR244 (1979), 1093–1096—Soviet Math. Dokl.20 (1979), 191–194.

    MathSciNet  Google Scholar 

  18. Khachyan L.G.,Polynomial algorithms in linear programming, Ž. Vyčisl. Mat. i Mat. Fiz.,20 (1980), 51–68, 260.

    Google Scholar 

  19. Kiefer J.C.,Collected papers 3:Design of experiments, Berlin, Springer 1984.

    Google Scholar 

  20. Kiefer J.C., Wolfwitz J.,The equivalence of two extrenum problems, Canad. J. Math.,12 (1960), 363–366.

    MATH  MathSciNet  Google Scholar 

  21. Koenig H., Pallaschke D.,On Khachian's algorithm and minimal ellipsoids, Numer. Math.,36 (1981), 211–223.

    Article  MATH  Google Scholar 

  22. Krafft O.,Dual optimization problems in stochastics, J.-ber. Deutsch. Math. Verein.83 (1981), 97–105.

    MATH  MathSciNet  Google Scholar 

  23. Laugwitz D.,Differentialgeometrie in Vektorräumen, Braunschweig, Vieweg 1965.

    MATH  Google Scholar 

  24. Pełczynski A.,Geometry of finite dimensional Banach spaces and operator ideals, In: H.E. Lacey, ed.: Notes in Banach spaces 81–181, Austin, Univ. of Texas Press 1980.

    Google Scholar 

  25. Roberts A.W., Varberg D.E.,Convex functions, New York, Academic Press 1973.

    MATH  Google Scholar 

  26. Rogers C.A.,Hausdorff measures, Cambridge, University Press 1970.

    MATH  Google Scholar 

  27. Ryškov S.S., Baranovskiî E.P.,Classical methods of the theory of lattice packings, Uspehi Mat. Nauk,34 (1979), 3–63, 256—Russ. Math. Surveys34 (1979), 1–68.

    Google Scholar 

  28. Schrader R.,Ellipsoidal methods, In: B. Korte, ed.: Modern applied mathematics (optimization and operations research) 256–311, Amsterdam, North-Holland 1982.

    Google Scholar 

  29. Sibson R.,Discussion of Dr. Wynn's and of Dr. Laycock's papers, J. Roy. Stat. Soc. (B)34 (1972), 181–183.

    Google Scholar 

  30. Silvey S.D.,Discussion of Dr. Wynn's and of Dr. Laycock's papers, J. Roy Stat. Soc. (B)34 (1972), 174–175.

    Google Scholar 

  31. Stoer J., Witzgall Chr.,Convexity and optimization in finite dimensions I, Berlin, Springer 1970.

    Google Scholar 

  32. Szarek S.J.,Volume estimates and nearly Euclidean decompositions for normed spaces, Sem. Funct. Analysis 1979–1980, École Polytechn. Palaiseau 1980, Exp. 25, 8pp.

  33. Titterington D.M.,Geometric approaches to the design of experiment, Math. Operationsforsch. Statistik. Ser.,11 (1980), 151–163.

    MATH  MathSciNet  Google Scholar 

  34. Zaguskin V.L.,On circumscribed and inscribed ellipsoids of extremal volume, Uspehi. Mat. Nauk.,13 (1958) (84), 89–93.

    MATH  MathSciNet  Google Scholar 

  35. Zamfirescu T.,Points on infinitely many normals to convex surfaces, J. reine angew. Math.350 (1984), 183–187.

    MATH  MathSciNet  Google Scholar 

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Gruber, P.M. Minimal ellipsoids and their duals. Rend. Circ. Mat. Palermo 37, 35–64 (1988). https://doi.org/10.1007/BF02844267

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