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Extremal functions for real convex bodies: simplices, strips, and ellipses

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Abstract

We present an explicit method to compute the (Siciak–Zaharjuta) extremal function of a real convex polytope in terms of supporting simplices and strips. We use this to give a new proof of the existence of extremal ellipses associated to the extremal function of a real convex body.

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Notes

  1. This terminology was introduced in [7].

  2. The definition that follows will also extend to degenerate ellipses; see Remark 7.5 below.

  3. To apply the lemma precisely, we also need the fact that some multiple of v translates a vertex of \(F_j\) into the interior of \(\Sigma \); this is easy to see.

  4. Recall that in terms of the c parameter, \({\mathcal {R}}_{K_0}(c\zeta _0)=z_0\).

  5. One can use the vertices of K to compute these (see e.g., [17]).

References

  1. Baran, M.: Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in \({\mathbb{R}}^n\). Michigan Math. J. 39(3), 395–404 (1992)

    Article  MathSciNet  Google Scholar 

  2. Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37, 1–44 (1976)

    Article  MathSciNet  Google Scholar 

  3. Bedford, E., Taylor, B.A.: Plurisubharmonic functions with logarithmic singularities. Ann. Inst. Fourier (Grenoble) 38(4), 133–171 (1988)

    Article  MathSciNet  Google Scholar 

  4. Bloom, T., Levenberg, N., Ma‘u, S.: Robin functions and extremal functions. Ann. Pol. Math., 80:55–84, (2003)

  5. Bos, L., Ma‘u, S., Waldron, S.: Extremal growth of polynomials. Anal. Math. 46(2), 195–224 (2020)

  6. Burns, D., Levenberg, N., Ma‘u, S.: Pluripotential theory for convex bodies in \({\mathbb{R}}^n\). Math. Z. 250, 91–111 (2005)

  7. Burns, D., Levenberg, N., Ma‘u, S.: Exterior Monge-Ampère solutions. Adv. Math. 222, 331–358 (2009)

  8. Burns, D., Levenberg, N., Ma‘u, S.: Extremal functions for real convex bodies. Ark. Mat. 53(2), 203–236 (2015)

  9. Burns, D., Levenberg, N., Ma‘u, S., Revesz, S.: Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities. Trans. Amer. Math. Soc. 362(12), 6325–6340 (2010)

  10. Hart, J., Ma‘u, S.: Chebyshev and Robin constants on algebraic curves. Ann. Polon. Math. 115(2), 101–121 (2015)

  11. Klimek, M.: Pluripotential Theory. The Clarendon Press, Oxford University Press, New York (1991)

    MATH  Google Scholar 

  12. Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France 109(4), 427–424 (1981)

    Article  MathSciNet  Google Scholar 

  13. Lempert, L.: Intrinsic distances and holomorphic retracts. In Complex analysis and applications ’81 (Varna, 1981), pages 341–364. Publ. House Bulgar. Acad. Sci., (1984)

  14. Lempert, L.: Symmetries and other transformations of the complex Monge-Ampère equation. Duke Math. J. 52(4), 869–885 (1985)

    Article  MathSciNet  Google Scholar 

  15. Levenberg, N., Perera, M.: A global domination principle for \(P\)-pluripotential theory. In Complex analysis and spectral theory, volume 743 of Contemp. Math., pages 11–19. Amer. Math. Soc., Providence, RI, (2020)

  16. Lundin, M.: The extremal PSH for the complement of convex, symmetric subsets of \({\mathbb{R}}^n\). Michigan Math. J. 32(2), 197–201 (1985)

    Article  MathSciNet  Google Scholar 

  17. Matt J. Analyze \(N\)-dimensional Polyhedra in terms of Vertices or (In)Equalities (https://www.mathworks.com/matlabcentral/fileexchange/30892-analyze-n-dimensional-polyhedra-in-terms-of-vertices-or-in-equalities). MATLAB Central File Exchange, (2020)

  18. Piazzon, F.: The extremal plurisubharmonic function of the torus. Dolomites Res. Notes Approx., 11(Special Issue Norm Levenberg):62–72, (2018)

  19. Sadullaev, A.: Estimates of polynomials on analytic sets. Izv. Akad. Nauk SSSR Ser. Mat. 46(3), 524–534 (1982)

    MathSciNet  MATH  Google Scholar 

  20. Trefethen, L.N.: Multivariate polynomial approximation in the hypercube. Proc. Amer. Math. Soc. 145(11), 4837–4844 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I would like to thank the referee for a number of helpful comments that simplified some of the arguments, and also for the formulation of Theorem 12.3.

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  1. University of Auckland, Auckland, New Zealand

    Sione Ma‘u

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  1. Sione Ma‘u
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Correspondence to Sione Ma‘u.

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Ma‘u, S. Extremal functions for real convex bodies: simplices, strips, and ellipses. Math. Z. 300, 3227–3267 (2022). https://doi.org/10.1007/s00209-021-02891-8

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