Geometric measure theory

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In mathematics, geometric measure theory (GMT) is the study of the geometric properties of the measures of sets (typically in Euclidean spaces), including such things as arc lengths and areas. It has applications in image processing.

Deep results in geometric measure theory identified a dichotomy between "rectifiable" or "regular sets" and measures on the one side, and non-rectifiable or fractal sets on the other.

Some basic results in geometric measure theory can turn out to have surprisingly far-reaching consequences. For example, the Brunn-Minkowski inequality for the n-dimensional volumes of convex bodies K and L,

$\mathrm{vol} \big( (1 - \lambda) K + \lambda L \big) \geq (1 - \lambda) \mathrm{vol} (K)^{1/n} + \lambda \mathrm{vol} (L)^{1/n},$

can be proved on a single page, yet quickly yields the classical isoperimetric inequality. The Brunn-Minkowski inequality also leads to Anderson's theorem in statistics. The proof of the Brunn-Minkowski inequality predates modern measure theory; the development of measure theory and Lebesgue integration allowed connections to be made between geometry and analysis, to the extent that in an integral form of the Brunn-Minkowski inequality known as the Prékopa-Leindler inequality the geometry seems almost entirely absent.

One application of geometric measure theory is the proof of Plateau's laws by Jean Taylor (building off work of Frederick J. Almgren, Jr.).