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For a given circle, is there exist an ellipse with same perimeter and area as to that circle?

If not, that is my suspicion, is in three-dimension parallel question: For a given sphere, is there exist an ellipsoid with same surface area and volume as to that sphere?

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  • $\begingroup$ I don't know a proof, but I would assume this is impossible. A circle is the shape which has the most area for the smallest perimeter. Similarly for a sphere. So, changing the shape and keeping the perimeter should decrease the area. $\endgroup$
    – N. Owad
    Commented May 17, 2016 at 21:16
  • $\begingroup$ Circle is an ellipse with $0$ eccentricity. So you are asking if exist two ellipses with the same area and perimeter while eccentricity is different. $\endgroup$
    – newzad
    Commented May 17, 2016 at 21:28

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No, because the isoperimetric quotient $A/P^2$ is smaller for a non-circular ellipse than for any circle.

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  • $\begingroup$ So bad, that is still a Chinese for me... $\endgroup$
    – Salech Alhasov
    Commented May 17, 2016 at 21:17
  • $\begingroup$ en.wikipedia.org/wiki/Isoperimetric_inequality . The circle is the unique curve for which the maximum value is attained. $\endgroup$
    – zyx
    Commented May 17, 2016 at 21:21
  • $\begingroup$ Amazing stuff! Thank you! $\endgroup$
    – Salech Alhasov
    Commented May 17, 2016 at 21:23
  • $\begingroup$ The answer in higher dimensions is the same, for the same reason. The pair (n-dimensional volume, (n-1)-dimensional surface area) is unique to the sphere in n dimensions and cannot be attained by any other shape. $\endgroup$
    – zyx
    Commented May 17, 2016 at 21:24
  • $\begingroup$ In 2 dimensions it is possible that Area and Perimeter determine the ellipse uniquely. A more difficult question. $\endgroup$
    – zyx
    Commented May 17, 2016 at 21:59
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No there isn't because the sphere is optimal in relation to the ratio of circumference to area. You can check this simply by writing the equation of the circumference divided by the area and differentiating with respect to the width to see that the minimum occurs when the width and length are equal.

Although technically a circle is an ellipse so the actual answer is yes, the circle.

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I cannot give you a deep rigourous argument, but surface tension is the reason a soap-bell is spherical so...

A circle has a specific eccentricity, and you choose another eccentricity while the area is fixed, there are no more parameters left to adjust for the perimeter.

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