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