Filling area conjecture
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points. == Definitions and statement of the conjecture == Every smooth surface M or curve in Euclidean space is a metric space, in which the (intrinsic) distance dM(x,y) between two points x, y of M is defined as the infimum of the lengths of the curves that go from x to y along M. For example, on a closed curve C {\displaystyle C} of length 2L, for each point x of the curve there is a unique other point of the curve (called the antipodal of x) at distance L from x.