Hadwiger's theorem

Hadwiger's theorem states that every continuous, rigid-motion-invariant valuation on convex bodies in Euclidean space is a linear combination of intrinsic volumes.

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Valuation (geometry): A function on geometric sets that is additive under unions when the intersection and union remain in its domain. Additivity is the defining algebraic condition imposed on functions in the theorem.
Steiner formula: A polynomial formula for the volume of a convex body enlarged by a Euclidean ball. Its coefficients are intrinsic volumes, linking parallel-body growth to the theorem's basis.
Hugo Hadwiger: A Swiss mathematician known for foundational work in convex geometry and integral geometry. The theorem bears his name and crystallized his classification of motion-invariant valuations.
Intrinsic-volume inequalities: Inequalities relating the intrinsic volumes of convex bodies, including the Alexandrov–Fenchel inequalities. The theorem reduces invariant valuations to quantities whose inequalities constrain geometric shape.
Hadwiger's characterization theorem for continuous valuations: A classification of continuous, translation-invariant valuations on convex bodies using homogeneous components. It broadens the classification beyond rigid-motion invariance, where more valuations are possible.
Convex body: A compact convex subset of Euclidean space with nonempty interior. These are the geometric objects on which the theorem's valuations are defined.
Parallel body: The Minkowski sum of a set with a ball of specified radius. Evaluating a valuation on these enlargements exposes the intrinsic-volume coefficients.
Blaschke's kinematic formulas: Integral-geometric formulas describing averages of geometric quantities under random rigid motions. Hadwiger's classification developed within the integral-geometric tradition these formulas shaped.
Brunn–Minkowski theory: The study of volume inequalities for convex sets under Minkowski addition. Its volume and mixed-volume methods interact directly with intrinsic volumes.
McMullen decomposition theorem: A theorem decomposing continuous translation-invariant valuations into homogeneous components. Its components clarify how imposing rotation invariance leads to Hadwiger's narrower classification.