Explanation
To explain the conjecture, we start with the observation that the equatorial circle of the unit 2-sphere
is a Riemannian circle S1 of length 2π and diameter π. More precisely, the Riemannian distance function of S1 is the restriction of the ambient Riemannian distance on the sphere. This property is not satisfied by the standard imbedding of the unit circle in the Euclidean plane, where a pair of opposite points are at distance 2, not π.
We consider all fillings of S1 by a surface, such that the restricted metric defined by the inclusion of the circle as the boundary of the surface is the Riemannian metric of a circle of length 2π. The inclusion of the circle as the boundary is then called a strongly isometric imbedding of the circle. In 1983 Gromov conjectured that the round hemisphere gives the "best" way of filling the circle among all filling surfaces.
Read more about this topic: Filling Area Conjecture
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