Zonoid

Template:Short description Template:Use mdy dates Template:Use list-defined references Template:CS1 config In convex geometry, a zonoid is a type of centrally symmetric convex body.

Definitions

The zonoids have several definitions, equivalent up to translations of the resulting shapes:Template:R

• 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.Template:R Approximating a zonoid to within Hausdorff distance $\epsilon$ requires a number of segments that (for fixed $\epsilon$) is near-linear in the dimension, or linear with some additional assumptions on the zonoid.Template:R
• 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.Template:R 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.Template:R
• 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.Template:R
• A zonoid is a convex set whose polar body is a projection body.Template:R

Examples

Every two-dimensional centrally-symmetric convex shape is a zonoid.Template:R In higher dimensions, the Euclidean unit ball is a zonoid.Template:R A polytope is a zonoid if and only if it is a zonotope.Template:R Thus, for instance, the regular octahedron is an example of a centrally symmetric convex shape that is not a zonoid.Template:R

The solid of revolution of the positive part of a sine curve is a zonoid, obtained as a limit of zonohedra whose generating segments are symmetric to each other with respect to rotations around a common axis.Template:R The bicones provide examples of centrally symmetric solids of revolution that are not zonoids.Template:R

Properties

Zonoids are closed under affine transformations,Template:R under parallel projection,Template:R and under finite Minkowski sums. Every zonoid that is not a line segment can be decomposed as a Minkowski sum of other zonoids that do not have the same shape as the given zonoid. (This means that they are not translates of homothetes of the given zonoid.)Template:R

The zonotopes can be characterized as polytopes having centrally-symmetric pairs of opposite faces, and the zonoid problem is the problem of finding an analogous characterization of zonoids. Ethan Bolker credits the formulation of this problem to a 1916 publication of Wilhelm Blaschke.Template:R

Further reading

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