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Existence and Uniqueness of a Curve with Both Minimal Length and Minimal Area

Author

Listed:
  • Ariel Fuxman

    (Department of Applied Mathematics, Holon Institute of Technology, Holon 5810201, Israel
    These authors contributed equally to this work.)

  • Shai Gul

    (Department of Applied Mathematics, Holon Institute of Technology, Holon 5810201, Israel
    These authors contributed equally to this work.)

Abstract

Consider the family of generalized parabolas { y = − a x r + c | a , r , c , x > 0 , r is a fixed constant } that pass through a given point in the first quadrant (and hence, depend on one parameter only). Find the parameter values for which the piece of the corresponding parabola in the first quadrant either encloses a minimum area, or has a minimum length. We find a sufficient condition under which given the fixed point, the area minimizing curve and the length minimizing curve coincide. The problem led us to a certain implicit function and we explored its asymptotic behavior and convexity.

Suggested Citation

  • Ariel Fuxman & Shai Gul, 2022. "Existence and Uniqueness of a Curve with Both Minimal Length and Minimal Area," Mathematics, MDPI, vol. 10(21), pages 1-18, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4061-:d:959870
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    References listed on IDEAS

    as
    1. Shai Gul & Reuven Cohen, 2021. "Efficient Covering of Thin Convex Domains Using Congruent Discs," Mathematics, MDPI, vol. 9(23), pages 1-10, November.
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    Keywords

    geometric optimization; implicit functions;

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