convex geometry In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of num ...

, a zonoid is a type of

centrally symmetric In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or ...

convex body In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non- empty interior. Some authors do not require a non-empty interior, merely that the set is non-empty. A convex body K is called symmetric if it ...

Definitions

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 In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets o ...

) by a

zonotope In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

, a convex polytope formed from the

Minkowski sum In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...

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 Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...

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 In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

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 Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...

L^1(,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(,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

Every two-dimensional centrally-symmetric convex shape is a zonoid. In higher dimensions, the Euclidean

is a zonoid. A polytope is a zonoid if and only if it is a zonotope. Thus, for instance, the

regular octahedron In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyh ...

is an example of a centrally symmetric convex shape that is not a zonoid. The

solid of revolution In geometry, a solid of revolution is a Solid geometry, solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution''), which may not Intersection (geometry), intersect the generatrix (except at its bound ...

of the positive part of a

sine curve A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is '' simple harmonic motion''; as rotation, it correspon ...

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. The

bicone In geometry, a bicone or dicone (from , and Greek: ''di-'', both meaning "two") is the three-dimensional surface of revolution of a rhombus around one of its axes of symmetry. Equivalently, a bicone is the surface created by joining two con ...

s provide examples of centrally symmetric solids of revolution that are not zonoids.

Properties

Zonoids are closed under

affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...

s, under

parallel projection In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the ''projection plane'' or ''image plane'', where the '' rays'', known as '' ...

, 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.) 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 Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taugh ...

Definitions

Examples

Properties

References

Further reading