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On the smoothness of the equizonal ovaloids

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Abstract

We show that the n-dimensional equizonal ovaloids are analytic when n is even and are of exactly C n-1 smoothness when n is odd. This substantially improves the previously published result on the smoothness of the even-dimensional equizonal ovaloids and slightly corrects the previously published statement regarding the smoothness of the odd-dimensional equizonal ovaloids. Our methods should be generally useful in determining the degree of smoothness of surfaces and hypersurfaces of revolution generated by piecewise-defined profile curves. In particular, they include a novel and elegant application of Bernstein’s theory of absolutely monotonic functions.

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Authors and Affiliations

  1. Department of Mathematics, Lehigh University, Christmas-Saucon Hall, 14 E. Packer Avenue, Bethlehem, PA, 18015, USA

    Vincent Coll

  2. Mathematical, Computing, and Information Sciences Department, Jacksonville State University, Jacksonville, AL, 36265, USA

    Jeff Dodd

  3. The Pennsylvania State University, 109A McAllister Building, University Park, PA, 16802, USA

    Michael Harrison

Authors
  1. Vincent Coll
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  2. Jeff Dodd
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  3. Michael Harrison
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Correspondence to Vincent Coll.

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Coll, V., Dodd, J. & Harrison, M. On the smoothness of the equizonal ovaloids. J. Geom. 103, 409–416 (2012). https://doi.org/10.1007/s00022-012-0135-1

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  • DOI: https://doi.org/10.1007/s00022-012-0135-1

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