Filling area conjecture

The filling area conjecture asserts that a hemisphere has the least area among disks spanning a circle when distances measured along the disk’s boundary are fixed.

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Filling area: The least area of a surface that spans a prescribed closed curve in a metric space. The conjecture proposes a specific value for this least area when the boundary is a metric circle.
Mikhail Gromov: A mathematician known for major contributions to geometry, topology, and geometric group theory. Gromov introduced the filling-area problem and formulated its central hemisphere bound.
Euclidean isoperimetric inequality: The inequality stating that a Euclidean sphere encloses the greatest volume for a given boundary area. It optimizes enclosed volume from boundary area, rather than spanning area from intrinsic boundary distances.
Filling area conjecture for the circle: The proposed sharp lower bound for disks whose boundary has the intrinsic metric of a circle. This is the canonical formulation, with the hemisphere as the candidate minimizer.
Intrinsic metric: A distance function defined by the lengths of paths constrained to lie within a space. The boundary constraint concerns distances measured within the disk, not through its ambient space.
Filling area inequality: A lower bound on the area required to fill a loop in terms of its length or metric geometry. The conjecture is the sharp circular-boundary case of this broader type of inequality.
Douglas–Rado theorem: The theorem asserting the existence of a minimal disk spanning a suitable Jordan curve in Euclidean space. It establishes existence of a least-area disk, not the conjectured hemisphere lower bound.
Filling area spectrum: The collection of least filling areas associated with prescribed boundary metric spaces. It asks how the hemisphere bound fits into the range of possible optimal filling values.
Hemisphere: One half of a sphere bounded by a great circle. It is the proposed area-minimizing disk for the prescribed circular boundary.
Filling radius: A measure of how far a space must be thickened before it can be filled in an ambient space. Gromov's related filling invariants place the area conjecture within a wider program.