The zonoids have several definitions, equivalent up to translations of the resulting shapes: • A zonoid is a shape that can be approximated arbitrarily closely (in
Hausdorff distance) by a
zonotope, a
convex polytope formed from the
Minkowski sum of finitely many line segments. In particular, every zonotope is a zonoid. Approximating a zonoid to within Hausdorff distance \varepsilon requires a number of segments that (for fixed \varepsilon) is near-linear in the dimension, or linear with some additional assumptions on the zonoid. • A zonoid is the
range of an atom-free vector-valued
sigma-additive set function. Here, a function from a family of sets to vectors is sigma-additive when the family is closed under countable disjoint unions, and when the value of the function on a union of sets equals the sum of its values on the sets. It is atom-free when every set whose function value is nonzero has a proper subset whose value remains nonzero. For this definition the resulting shapes contain the origin, but they may be translated arbitrarily as long as they contain the origin. The statement that the shapes described in this way are closed and convex is known as
Lyapunov's theorem. • A zonoid is the
convex hull of the range of a vector-valued
sigma-additive set function. For this definition, being atom-free is not required. • A zonoid is the
polar body of a central section of the
unit ball of L^1([0,1]), the
space of Lebesgue integrable functions on the unit interval. Here, a central section is the intersection of this ball with a finite-dimensional subspace of L^1([0,1]). This definition produces zonoids whose center of symmetry is at the origin. • A zonoid is a
convex set whose polar body is a
projection body. ==Examples==