European Journal of Mathematics

, Volume 5, Issue 3, pp 1013–1032 | Cite as

Relating transfinite diameters using an Okounkov body

  • Sione Ma‘uEmail author
Research Article


We derive relations between the transfinite diameter of a locally circled subset K of the complexified sphere in \(\mathbb {C}^3\) and notions of weighted transfinite diameter of the projection of K to \(\mathbb {C}^2\). Our method is based on connecting a Chebyshev transform of K on an Okounkov body of V to the classical Chebyshev transform of the projection of K to \(\mathbb {C}^2\).


Transfinite diameter Okounkov body Chebyshev transform Affine variety 

Mathematics Subject Classification

32U20 14M25 52B20 



  1. 1.
    Berman, R., Boucksom, S.: Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math. 181(2), 337–394 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bloom, T., Bos, L., Christensen, C., Levenberg, N.: Polynomial interpolation of holomorphic functions in \(\mathbf{C}\) and \({\mathbf{C}}^n\). Rocky Mountain J. Math. 22(2), 441–470 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bloom, T., Bos, L.P., Calvi, J.-P., Levenberg, N.: Polynomial interpolation and approximation in \({\mathbf{C}}^d\). Ann. Polon. Math. 106, 53–81 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bloom, T., Levenberg, N.: Weighted pluripotential theory in \({\mathbf{C}}^N\). Amer. J. Math. 125(1), 57–103 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boucksom, S., Chen, H.: Okounkov bodies of filtered linear series. Compositio Math. 147(4), 1205–1229 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cox, D.A., Ma‘u, S.: Transfinite diameter on complex algebraic varieties. Pacific J. Math. 291(2), 279–317 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    DeMarco, L., Rumely, R.: Transfinite diameter and the resultant. J. Reine Angew. Math. 611, 145–161 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kaveh, K., Khovanskii, A.G.: Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. 176(2), 925–978 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Küronya, A., Lozovanu, V., Maclean, C.: Convex bodies appearing as Okounkov bodies of divisors. Adv. Math. 229(5), 2622–2639 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. 42(5), 783–835 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ma‘u, S.: Newton–Okounkov bodies and transfinite diameter. Dolomites Res. Notes Approx. 10(Special Issue), 138–160 (2017). arXiv:1701.00570 MathSciNetzbMATHGoogle Scholar
  12. 12.
    Witt Nyström, D.: Transforming metrics on a line bundle to the Okounkov body. Ann. Sci. Éc. Norm. Supér. 47(4), 1111–1161 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zaharjuta, V.P.: Transfinite diameter, Čebyšev constants, and capacity for compacta in \(\mathbb{C}^n\). Sb. Math. 25(3), 350–364 (1975)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand

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