Geometry of good sets inn-fold Cartesian product

  • A. Kłopotowski
  • M. G. Nadkarni
  • K. P. S. Bhaskara Rao


We propose here a multidimensional generalisation of the notion of link introduced in our previous papers and we discuss some consequences for simplicial measures and sums of function algebras.


Good set full sets geodesics boundary 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Beneš V and ⋆epán J, The support of extremal probability measures with given marginals, mathematical statistics and probability theory (eds) M L Puriet al (1987) (D Reidel Publishing Company) vol. A, pp. 33–41Google Scholar
  2. [2]
    Beneš V and štepán J, Extremal solutions in the marginal problem, in: Advances in probability distributions with given marginals (Dordrecht: Kluwer Academic Publishers) (1991) pp. 189–207Google Scholar
  3. [3]
    Cowsik R C, Klopotowski A and Nadkarni M G, When is ƒ(x, y) =u(x) +v(y)?,Proc. Indian Acad. Sci. (Math. Sci.) 109 (1999) 57–64MATHMathSciNetGoogle Scholar
  4. [4]
    Douglas R G, On extremal measures and subspace density,Michigan Math. J. 11 (1964) 243–246MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Hestir K and Williams S C, Supports of doubly stochastic measures,Bernoulli 1(3) (1995) 217–243MATHMathSciNetGoogle Scholar
  6. [6]
    Kłopotowski A, Nadkarni M G, Sarbadhikari H and Srivastava S M, Sets with doubleton sections, good sets and ergodic theory,Fund. Math. 173 (2002) 133–158MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Kłopotowski A, Nadkarni M G and Bhaskara Rao K P S, When is ƒ (x 1,X 2,…,x n) =u 1(x 1) +u(x 2) + … + (u (n(x (n)?Proc. Indian Acad, Sci. (Math. Sci.) 113 (2003) 77–86MathSciNetMATHGoogle Scholar
  8. [8]
    Kolmogorov A N, On the representation of continuous functions of several variables a superposition of continuous functions of one variable and addition,Dokl. Acad. Nauk. SSSR 114 (1957) 679–681;Am. Math. Soc. Transl. 28 (1963) 55–59MathSciNetGoogle Scholar
  9. [9]
    Lindenstrauss J, A remark on doubly stochastic measures,Am. Math. Monthly 72 (1965) 379–382MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Marshall D E and O’Farrell A G, Uniform approximation by real functions,Fund. Math. CIV (1979) 203–211MathSciNetGoogle Scholar
  11. [11]
    Marshall D E and O’Farrell A G, Approximation by sums of two algebras, the lightening bolt principle,J. Fund. Anal. 52 (1983) 353–368MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Medvedev V A, On the sum of two closed algebras of continuous functions on a compactum,Funktsionaĺnyi Analiz i Ego Prilozheniya,27 (1993) 33–36 (English Translation: Plenum Publishing Corp. 1993)MATHGoogle Scholar
  13. [13]
    Mehta R D and Vasavada M H, Algebra direct sum decomposition of CR(X),Proc. Am. Math. Soc. 98 (1986) 71–74MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Mehta R D and Vasavada M H, Algebra direct sum decomposition of Cr(X), II,Proc. Am. Math. Soc. 100 (1987) 123–126MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Navada K G, Some remarks on good sets (Preprint)Google Scholar
  16. [16]
    Sproston J P and Strauss D, Sums of subalgebras of C(X),J. London Math. Soc. 2(45) (1992) 265–278CrossRefMathSciNetGoogle Scholar
  17. [17]
    štepán J, Simplicial measures and sets of uniqueness in the marginal problem, in: Statistics and decision 11 (München: R Oldenbourg Verlag) (1993) pp. 289–299Google Scholar
  18. [18]
    Sternfeld Y, Uniformly separating families of functions,Israel J. Math. 29 (1978) 61–91MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Sternfeld Y, Uniform separation of points and measures and representation by sums of algebras,Israel J. Math. 55 (1986) 350–362MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2004

Authors and Affiliations

  • A. Kłopotowski
    • 1
  • M. G. Nadkarni
    • 2
    • 3
  • K. P. S. Bhaskara Rao
    • 4
  1. 1.Institut GaliléeUniversité Paris XIIIVilletaneuse CedexFrance
  2. 2.Institute of Mathematical SciencesChennaiIndia
  3. 3.Chennai Mathematical InstituteChennaiIndia
  4. 4.Department of MathematicsSouthwestern CollegeWinfieldUSA

Personalised recommendations