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More on the Potential for the Farthest-Point Distance Function

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Abstract

For some particular cases in dimension 3 and higher we prove a conjecture of Laugesen and Pritsker (Canad. Math. Bull., 46, pp. 373–387, 2003) concerning the probability measure related with the farthest-point distance function. In dimension 2 we offer an alternative proof for the result by Gardiner and Netuka. We also consider some examples that may provide more insight in the nature of the problem.

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References

  1. Anzelotti, G.: Pairings between measures and functions and compensated compactness. Ann. Mat. Pura. Appl. 135, 293–318 (1983)

    Article  MathSciNet  Google Scholar 

  2. Armitage, D.H., Gardiner, S.J.: Classical potential theory, Springer Monographs in Mathematics. Springer , London (2001). Ltd., London

    Book  Google Scholar 

  3. Blaschke, W.: Einige Bemerkungen über Kurven und Flächen von konstanter Breite. Ber. Verh. Sächs. Akad. Leipzig 67, 290–297 (1915)

    Google Scholar 

  4. Boyd, D.W: Sharp inequalities for the product of polynomials. Bull. London Math. Soc. 26, 449–454 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chakerian, G.D., Groemer, H.: Convex bodies of constant width. In: Gruber, P. M., Wills, J. M. (eds.) Convexity and its Applications, Birkhäuser, Basel, pp. 4–96 (1983)

  6. Chen, G.-Q., Frid, H.: On the theory of divergence-measure fields and its applications. Bol. Soc. Brasil. Mat. 32, 401–433 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gardiner, S.J., Netuka, I.: Potential theory of the farthest point distance function. J. d’Analyse Math. 101, 163–177 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gardiner, S.J., Netuka, I.: The Farthest Point Distance Function. In: Complex and Harmonic Analysis, Carberym, A. et al. (Eds), DEStech Publications Inc., Lancaster, Pennsylvania, USA pp. 35–43 (2007)

  9. Hernández Cifre, M.A., Salinas, G., Gomis, S.S.: Two optimization problems for convex bodies in the n-dimensional space. Beiträge Algebra Geom. 45, 549–555 (2004)

    MATH  Google Scholar 

  10. Kawohl, B.: Convex sets of constant width. Oberwolfach Rep. 6, 390–393 (2009)

    Google Scholar 

  11. Kawohl, B., Weber, C.: Meissner’s mysterious bodies. Math. Intelligencer 33, 94–101 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Laugesen, R., Pritsker, I.E.: Potential Theory of the Farthest-Point Distance Function. Canad. Math. Bull. 46, 373–387 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Pritsker, I.E.: Products of polynomials in uniform norms. Trans. Amer. Math. Soc. 353, 3971–3993 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pritsker, I.E., Saff, E.B., Wise, W.: Reverse triangle inequalities for Riesz potentials and connections with polarization. J. Math. Anal. Appl. 410, 868–881 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  15. Rockafellar, R.T.: Convex analysis, Princeton Mathematical . No. 28 . Princeton University Press, Princeton (1970)

  16. Schneider, R.: Convex bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  17. Wise, W.: Potential theory and geometry of the farthest distance function. Potential Anal. 39, 341–353 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wise, W.: Potential Theory of the farthest distance function, PhD Thesis. Oklahoma State University, Stillwater OK (2014)

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Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, D 50923, Köln, Germany

    Bernd Kawohl, Carlo Nitsch & Guido Sweers

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  1. Bernd Kawohl
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  2. Carlo Nitsch
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  3. Guido Sweers
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Correspondence to Guido Sweers.

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Kawohl, B., Nitsch, C. & Sweers, G. More on the Potential for the Farthest-Point Distance Function. Potential Anal 42, 699–716 (2015). https://doi.org/10.1007/s11118-014-9454-1

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  • DOI: https://doi.org/10.1007/s11118-014-9454-1

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