Geometric Measure Theory

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6 domain-specific abstractions whose origin domain is Geometric Measure Theory.

• Caccioppoli set — A measurable set whose characteristic function has locally bounded variation, equivalently a set of locally finite perimeter in the geometric-measure-theory sense.
• Flat convergence — Convergence of geometric chains or currents in the flat norm, permitting their difference to be decomposed into a small-mass current plus the boundary of another small-mass current.
• Hausdorff density — The small-scale upper, lower or exact ratio of a Radon measure's mass in balls around a point to the radius raised to a declared dimension.
• Kakeya Set — A subset of Euclidean space containing a unit line segment in every direction, whose directional coverage can coexist with vanishing measure and extreme geometric overlap.
• Tangent measure — A weak limit of rescaled blow-ups of a Radon measure around a point, capturing its infinitesimal mass geometry.
• Varifold — A Radon measure on position–tangent-plane space representing a generalized surface with mass and orientation-free tangent information.