Geometric Measure Theory

Two well-known mathematical problems are considered to be the precursors of geometric measurement theory: the isoperimetric problem and the Plateau problem.

The isoperimetric problem (or the problem of Dido, legendary founder and first queen of Carthage) consists in determining a plane figure with the largest possible area, but whose boundary has a specified length. This problem can be generalized to surfaces in three-dimensional Euclidean space, and also to hypersurfaces in dimensions greater than 3. The solution in the plane (i.e. a disk) was already known in ancient Greece (by Zenodorus), but the first mathematically rigorous proof was only obtained in the 19th century.

Plateau's (1849) problem (posed by J.-L. Lagrange in 1760) consists in determining a minimal surface (i.e. a minimal area) resting on a given closed edge. This problem is generalized to dimensions greater than 3. General solutions in the Euclidean plane were first published in the early 1930s, and in 1961 in dimensions greater than 3.

A. Besicovitch pioneered geometric measure theory in the Euclidean plane in the late 1920s and 1930s. In the 1950s and 1960s, H. Federer extended A. Besicovitch's work to Euclidean spaces of dimension greater than 2, and published a major treatise on this theory in 1969, of which he is considered the principal "founder".