Zonoid

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:{{r|b69}}

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.{{r|b69}} 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.{{r|blm}}
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.{{r|b69}} 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.{{r|b69}}
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.{{r|b69}}
A zonoid is a convex set whose polar body is a projection body.{{r|b69}}

Examples

Every two-dimensional centrally-symmetric convex shape is a zonoid.{{r|b71}} In higher dimensions, the Euclidean unit ball is a zonoid.{{r|b69}} A polytope is a zonoid if and only if it is a zonotope.{{r|blm}} Thus, for instance, the regular octahedron is an example of a centrally symmetric convex shape that is not a zonoid.{{r|b69}}

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.{{r|cc}} The bicones provide examples of centrally symmetric solids of revolution that are not zonoids.{{r|b69}}

Properties

Zonoids are closed under affine transformations,{{r|blm}} under parallel projection,{{r|rz}} 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.){{r|b69}}

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.{{r|b71}}

References

{{reflist|refs=

{{citation

| last = Bolker | first = Ethan D.

| doi = 10.2307/1995073

| journal = Transactions of the American Mathematical Society

| mr = 256265

| pages = 323–345

| title = A class of convex bodies

| volume = 145

| year = 1969| jstor = 1995073

}}

{{citation

| last = Bolker | first = E. D.

| department = Research Problems

| doi = 10.2307/2317764

| issue = 5

| journal = The American Mathematical Monthly

| jstor = 2317764

| mr = 1536334

| pages = 529–531

| title = The zonoid problem

| volume = 78

| year = 1971}}

{{citation

| last1 = Bourgain | first1 = J. | author1-link = Jean Bourgain

| last2 = Lindenstrauss | first2 = J. | author2-link = Joram Lindenstrauss

| last3 = Milman | first3 = V. | author3-link = Vitali Milman

| doi = 10.1007/BF02392835

| issue = 1–2

| journal = Acta Mathematica

| mr = 981200

| pages = 73–141

| title = Approximation of zonoids by zonotopes

| volume = 162

| year = 1989}}

{{citation

| last1 = Chilton | first1 = B. L.

| last2 = Coxeter | first2 = H. S. M. | author2-link = Harold Scott MacDonald Coxeter

| doi = 10.2307/2313051

| journal = The American Mathematical Monthly

| jstor = 2313051

| mr = 157282

| pages = 946–951

| title = Polar zonohedra

| volume = 70

| year = 1963| issue = 9

}}

{{citation

| last1 = Ryabogin | first1 = Dmitry

| last2 = Zvavitch | first2 = Artem

| contribution = Analytic methods in convex geometry

| contribution-url = https://www.impan.pl/dzialalnosc/studia-doktoranckie/studia/special-lectures-for-phd-students/rz2011.pdf

| isbn = 978-83-86806-24-9

| mr = 3329057

| pages = 87–183

| publisher = Polish Acad. Sci. Inst. Math., Warsaw

| series = IMPAN Lect. Notes

| title = Analytical and probabilistic methods in the geometry of convex bodies

| volume = 2

| year = 2014}}; see in particular section 4, "Zonoids and zonotopes"

}}

Zonoid

Definitions

Examples

Properties

References

Further reading