Convex body

A convex body is a compact convex subset of Euclidean space with nonempty interior. It may have a smooth or nonsmooth boundary and need not be strictly convex.

Convex set: A set containing the line segment between every pair of its points. Convexity is the defining condition inherited by every convex body.
Support function: A function giving the maximum value of a linear functional over a set in each direction. It records the furthest extent of a convex body along every direction.
Brunn–Minkowski inequality: An inequality relating the volumes of two sets to the volume of their Minkowski sum. It gives a fundamental constraint on how volume changes when bodies are added.
Convex cone: A convex set closed under addition and multiplication by nonnegative scalars. Unlike a convex body, a nontrivial cone is unbounded and has no compact interior as a whole.
Hermann Minkowski: A German mathematician whose work shaped number theory, geometry, and spacetime physics. His geometry of numbers and addition of convex sets helped found modern convex-body theory.
Interior (topology): The largest open set contained in a given topological space or subset. A convex body must contain an open region, not merely lie in a lower-dimensional subspace.
Minkowski functional: A function measuring how far a point lies from the origin relative to a convex absorbing set. For a body containing the origin, it encodes the body through its scaled copies.
Isoperimetric inequality: An inequality stating that a ball maximizes volume among sets with fixed surface area. Convex bodies provide a central setting for sharp volume–surface comparisons.
Polytope: The convex hull of finitely many points in Euclidean space. Polytopes are a discrete, face-rich subclass of convex bodies.
Hermann Brunn: A German mathematician known for foundational work on convex geometry and volume inequalities. His work with Minkowski led to the inequality governing volumes of sums of bodies.

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