Monatshefte für Mathematik

, Volume 145, Issue 4, pp 307–319 | Cite as

Sylvester’s Problem for Symmetric Convex Bodies and Related Problems

  • Mark W. Meckes
Article
  • 65 Downloads

Abstract.

We consider moments of the normalized volume of a symmetric or nonsymmetric random polytope in a fixed symmetric convex body. We investigate for which bodies these moments are extremized, and calculate exact values in some of the extreme cases. We show that these moments are maximized among planar convex bodies by parallelograms.

2000 Mathematics Subject Classifications: 52A40; 52A22, 60D05 
Key words: Sylvester’s problem, volumes of random polytopes, linear parameter systems 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Affentranger, F 1988The expected volume of a random polytope in a ball.J Microscopy151277287Google Scholar
  2. Alagar, VS 1977On the distribution of a random triangle.J Appl Probability14284297Google Scholar
  3. Ball, K 1988Logarithmically concave functions and sections of convex sets in \({\Bbb R}^n\).Studia Math886984Google Scholar
  4. Bárány, I, Buchta, C 1993Random polytopes in a convex polytope, independence of shape, and concentration of vertices.Math Ann297467497CrossRefGoogle Scholar
  5. Bisztriczky, T, Böröczky, K,Jr 2003About the centroid body and the ellipsoid of inertia.Mathematika48113Google Scholar
  6. Blaschke, W 1917Über affine Geometrie XI: Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten.Ber Verh Sächs Akad Leipzig69436453Google Scholar
  7. Blaschke W (1923) Vorlesungen über Differentialgeometrie II: Affine Differentialgeometrie. Berlin: SpringerGoogle Scholar
  8. Bourgain J (1991) On the distribution of polynomials on high-dimensional convex sets. In Geometric Aspects of Functional Analysis. Lect Notes Math 1469: pp 127–137. Berlin Heidelberg New York: SpringerGoogle Scholar
  9. Buchta, C 1984Zufallspolygone in konvexen Vielecken.J Reine Angew Math347212220Google Scholar
  10. Buchta, C, Reitzner, M 1997On a theorem of G. Herglotz about random polygons.Suppl Rend Circ Mat Palermo, II Ser5089102Google Scholar
  11. Buchta, C, Reitzner, M 2001The convex hull of random points in a tetrahedron: solution of Blaschke’s problem and more general results.J Reine Angew Math536129Google Scholar
  12. Campi, S, Colesanti, A, Gronchi, P 1999A note on Sylvester’s problem for random polytopes in a convex body.Rend Istit Mat Univ Trieste317994Google Scholar
  13. Dalla L, Larman DG (1991) Volumes of a random polytope in a convex set. In Applied Geometry and Discrete Mathematics, pp 175–180. Providence, RI: Amer Math SocGoogle Scholar
  14. Efron, B 1965The convex hull of a random set of points.Biometrika52331343Google Scholar
  15. Giannopoulos, AA 1992On the mean value of the area of a random polygon in a plane convex body.Mathematika39279290Google Scholar
  16. Groemer, H 1973On some mean values associated with a randomly selected simplex in a convex set.Pacific J Math45525533Google Scholar
  17. Groemer, H 1974On the mean value of the volume of a random polytope in a convex set.Arch Math (Basel)258690Google Scholar
  18. Henze, N 1983Random triangles in convex regions.J Appl Probab20111125Google Scholar
  19. Kingman, JFC 1969Random secants of a convex body.J Appl Probab6660672Google Scholar
  20. Meckes, MW 2004Volumes of symmetric random polytopes.Arch Math (Basel)828596Google Scholar
  21. Miles, RE 1971Isotropic random simplices.Advances Appl Probab3353382Google Scholar
  22. Milman VD, Pajor A (1989) Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In Geometric Aspects of Functional Analysis. Lect Notes Math 1376: pp 64–104. Berlin Heidelberg New York: SpringerGoogle Scholar
  23. Paouris G (2000) On the isotropic constant of non-symmetric convex bodies. In Geometric Aspects of Functional Analysis. Lect Notes Math 1745: pp 239–243. Berlin Heidelberg New York: SpringerGoogle Scholar
  24. Pfiefer, RE 1989The historical development of J. J. Sylvester’s four point problem.Math Mag62309317Google Scholar
  25. Rényi, A, Sulanke, R 1963Über die konvexe Hülle von n zufällig gewählten Punkten.Z Wahrscheinlichkeitstheorie Verw Gebiete27584CrossRefGoogle Scholar
  26. Rényi, A, Sulanke, R 1964Über die konvexe Hülle von n zufällig gewählten Punkten. II.Z Wahrscheinlichkeitstheorie Verw Gebiete3138147CrossRefGoogle Scholar
  27. Rogers, CA, Shephard, GC 1958Some extremal problems for convex bodies.Mathematika593102Google Scholar
  28. Santaló LA (1976) Integral Geometry and Geometric Probability, vol 1 of Encyclopedia of Mathematics and its Applications. Reading, Mass: Addison-WesleyGoogle Scholar
  29. Schmuckenschläger, M 1998Volume of intersections and sections of the unit ball of l p n .Proc Amer Math Soc12615271530CrossRefGoogle Scholar
  30. Schneider R, Weil W (1992) Integralgeometrie. Stuttgart: TeubnerGoogle Scholar
  31. Zinani, A 2003The expected volume of a tetrahedron whose vertices are chosen at random in the interior of a cube.Monatsh Math139341348CrossRefGoogle Scholar

Copyright information

© Springer-Verlag/Wien 2005

Authors and Affiliations

  • Mark W. Meckes
    • 1
  1. 1.Stanford UniversityStanfordUSA

Personalised recommendations