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Analytic Parametrization of Three-Dimensional Bodies of Constant Width

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Abstract

We present a complete analytic parametrization of constant three-dimensional width bodies based on the median surface: more precisely, we define a bijection between spaces of functions and constant width bodies. We compute simple geometrical quantities like the volume and the surface area in terms of those functions. As a corollary we give a new algebraic proof of Blaschke’s formula. Finally, we derive weak optimality conditions for convex bodies which minimize the volume among constant width bodies.

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Authors and Affiliations

  1. Laboratoire de mathématiques, Université de Savoie, Campus scientifique, 73376, Le Bourget-du-lac, France

    T. Bayen, T. Lachand-Robert & É. Oudet

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  1. T. Bayen
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  2. T. Lachand-Robert
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  3. É. Oudet
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Correspondence to T. Bayen.

Additional information

Communicated by G. Dal Maso

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Bayen, T., Lachand-Robert, T. & Oudet, É. Analytic Parametrization of Three-Dimensional Bodies of Constant Width. Arch Rational Mech Anal 186, 225–249 (2007). https://doi.org/10.1007/s00205-007-0060-x

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  • DOI: https://doi.org/10.1007/s00205-007-0060-x

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